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PARABOLA, , PARABOLA, 1. DEFINITION, A parabola is the locus of a point which moves in a plane such that its distance from a fixed point (called, the focus) is equal to its distance from a fixed straight line (called the directrix)., Y, N, , P(x, y), y2 = 4ax, , L, , Q, , A, , S(a, 0), , X, , L', , Let S be the focus. QN be the directrix and P be any point on the parabola. Then by definition. PS = PN where, PN is the length of the perpendicular from P on the directrix QN., , 2. TERMS RELATED TO PARABOLA, Axis : A straight line passes through the focus and perpendicular to the directrix is called the axis of parabola., Vertex : The point of intersection of a parabola and its axis is called the vertex of the parabola., The vertex is the middle point of the focus and the point of intersection of axis and directrix., Eccentricity : If P be a point on the parabola and PN and PS are the distance from the directrix and focus, S respectively then the ratio PS/PN is called the eccentricity of the parabola which is denoted by e., By the definition for the parabola e = 1., If e > 1   , , , , hyperbola, e = 0, , y, , N, , circle, e < 1  ellipse, L, , Foc, , L(a, 2a), P, x=a, , Double ordinate, , is, al d, , Directrix, , tanc, e, , Vertex, A, , x+a=0, , Q, , Focal chord, , Focus, S (a, 0), , axis, , x, , Latus Rectum, L', y', , L' (a, -2a), , Latus Rectum, Let the given parabola be y2 = 4ax. In the figure LSL' (a line through focus  to axis) is the latus rectum., Also by definition,, LSL' = 2 ( 4a.a )  4a, = double ordinate (Any chord of the parabola y2 = 4ax which is  to its axis is called the double ordinate), through the focus S., Note : Two parabolas are said to be equal when their latus recta are equal., Focal Chord, Any chord to the parabola which passes through the focus is called a focal chord of the parabola., 90

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PARABOLA, , 3. FOUR STANDARD FORMS OF THE PARABOLA, y2 = 4ax (a>0), , Shape of Parabola, , S(a, 0), L', , x, , x2 = 4ay(a>0), Y, , Y, , P(x, y), , L, A, (0,0), , y2 = –4ax(a>0), , L P(x, y), , 2, , x=0 x=a, , y=0, , L, P(x, y), , X, , 2, , x=a x=0, , x2 =–4ay(a > 0), y=a, , A, y=0, (0,0), , L', y = -4ax, , y = 4ax, , L', , A, (0,0), , S(-a, 0), , S(0, a), , Standard Equation, , 2, , x = 4ay, , A(0,0), , P(x, y), X, , L', , y = -a, , L, S(0, -a), , x2 = -4ay, , Vertex, , A(0, 0), , A(0, 0), , A(0, 0), , A(0, 0), , Focus, , S(a, 0), , S(–a, 0), , S(0, a), , S(0, –a), , Equation of directrix, , x = –a, , x = a, , y = –a, , y = a, , Equation of axis, , y = 0, , y = 0, , x = 0, , x = 0, , Length of latus rectum, , 4a, , 4a, , 4a, , 4a, , Extermities of latus, , (a, ±2a), , (–a, ±2a), , (±2a, a), , (±2a, –a), , x = a, , x = –a, , y = a, , y = –a, , x = 0, , x = 0, , y = 0, , y = 0, , x + a, , x – a, , y + a, , y – a, , Parametric coordinates, , (at2, 2at), , (–at2, 2at), , (2at, at2), , (2at, –at2), , Eccentricity (e), , 1, , 1, , 1, , 1, , rectum, Equation of latus, rectum, Equation of tangents, at vertex, Focal distance of a, point P(x, y), , Ex.1, , Find the equation of the parabola whose vertex is (–3, 0) and directrix is x + 5 = 0., , Sol., , A line passing through the vertex (–3, 0) and perpendicular to directrix x + 5 = 0 is the axis of the parabola. Let, focus of the parabola is (a, 0). Since vertex, is the middle point of (–5, 0) and focus S, therefore, , 3 , , , (a  5 ), 2, ,  a = –1, , Focus = (–1, 0), , Thus the equation to the parabola is, (x + 1)2 + y2 = (x + 5)2, , , , y2 = 8(x + 3), , 4. REDUCTION OF STANDARD EQUATION, If the equation of a parabola contains second degree term either in y or in x(but not in both) then it can be, reduced into standard form. For this we change the given equation into the following forms(y – k)2 = 4a (x – h) or (x – p)2 = 4b (y – q), 91

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PARABOLA, , Then we compare from the following table for the results related to parabola., Equation of Parabola Vertex, , Axis, , Focus, , (y–k)2 = 4a(x – h), , (h, k), , y = k, , (h + a, k), , x + a – h = 0, , x = a + h, , 4a, , (x–p)2 = 4b(y – q), , (p, q), , x = p, , (p, b + q), , y + b – q = 0, , y = b + q, , 4b, , Ex.2, , Find the vertex of the parabola x2 – 8y – x + 19 = 0., , Sol., , Given equation can be written as, , Directrix, , Equation of L.R. Length of L.R., , 2, , 1, 1, ,  x    8 y  19   0, 2, 4, , 2, , , , 1, 76  1, ,  x  2   8y  4, , , , 2, , 1, 75 , , , ,   x    8 y , 2, 32 , , , ,  1 75 , , vertex =  ,,  2 32 , , , , 5. GENERAL EQUATION OF A PARABOLA, If (h, k) be the locus of a parabola and the euqation of directrix is ax + by + c = 0, then its equation is given, by, , ( x  h)2  ( y  k )2 , , ax  by  c, , a2  b2, where g, f, d are the constant., , which gives (bx – ay)2 + 2gx + 2fy + d = 0, , Note, , , The general equation of second degree ax2 + by2 + 2hxy + 2gx + 2fy + c = 0 represents a parabola, if, (a) h2 = ab, (b)  = abc + 2fgh – af2 – bg2 – ch2  0, , 6. PARAMETRIC EQUATIONS OF A PARABOLA, Clearly x = at2, y = 2at satisfy the equation y2 = 4ax for all real values of t. Hence the parametric equation, of the parabola y2 = 4ax are x = at2, y = 2at, where t is the parameter., Also, (at2, 2at) is a point on the parabola y2 = 4ax for all real values of t. This point is also described as the, point 't' on the parabola., Ex.3, , Find the parameter t of a point (4, –6) of the parabola y2 = 9x., , Sol., , Parametric coordinates of any point on parabola y2 = 4ax are, (at2, 2at), Here, , 4a = 9  a = 9/4, , since, , y coordinates is = –6, , , , 2(9/4) t = –6  t = –4/3, 92

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PARABOLA, , Ex.5, , If the line 2x – 3y = k touches the parabola y2 = 6x, find the value of k., , Sol., , Given, , x, , and, , y2 = 6x, , 3y  k, 2, , .....(i), .....(ii), , , ,  3y  k , y2  6 , ,  2 , , , , y2 = 3(3y + k)  y2 – 9y – 3k = 0, , .....(iii), , If line (i) touches parabola (ii) then roots of quadratic equation (iii) is equal, (–9)2 = 4 × 1 × (–3k)  k = , , so, , 27, 4, , 9. EQUATION OF TANGENT IN DIFFERENT FORMS, (i) Point Form, The equation of the tangent to the parabola y2 = 4ax at the point (x1, y1) is, yy1 = 2a (x + x1), Note :, The equation of the tangent at (x1, y1) can also obtained by replacing x2 by xx1, y2 by yy1, x by, , x  x1, ,, 2, , xy1  x1y, y  y1, and xy by, . This method is used only when the equation of parabola is polynomial of, 2, 2, second degree in x and y., , y by, , (ii) Parametric Form, The equation of the tangent to the parabola y2 = 4ax at the point (at2, 2at) is, ty = x + at2., (iii) Slope Form, The equation of tangent to the parabola y2 = 4ax in terms of slope 'm' is, a, ., y = mx +, m,  a 2a , , The coordinate of the point of contact are  2 ,, m m , Ex.6, Find the slope of tangent lines drawn from (3, 8) to the parabola y2 = –12x., Sol., , Since 82 + 12 × 3  0, therefore the point (3, 8) is not on the parabola, Now equation of any tangent to the parabola y2 = –12x is written as, 3, y = mx –  , m, , Since this line passes through (3, 8), so, , 3, 8 = 3m –  , m, , Solving this equation, we get m = 3, , ,  3m2 – 8m – 3 = 0, , 1, 3, 94

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PARABOLA, , 10. POINT OF INTERSECTION OF TANGENTS, The point of intersection of tangents drawn at two different points of contact P(at12, 2at1) and Q(at22, 2at2) on, the parabola y2 = 4ax is, , R, , slo, , =, pe, , 1/ t 1, 2, P(at1 , 2at1), , Sl, O, op, 2, e= Q (at2 , 2at 2), 1/, t2, , X, , R=(at1t2, a(t1 + t2))., , Note :, , , Angle between tangents at two points P(at12, 2at1) and Q(at22, 2at2) on the parabola y2 = 4ax is, t t,   tan1 2 1, 1  t1t2, , , , The G.M. of the x-coordinates of P and Q (i.e.,, , at 12  at 22 = at1t2) is the x-coordinate of the point of, , intersection of tangents at P and Q on the parabola., , , 2at 1  2at 2, , ,  a( t1  t 2 )  is the y-coordinate of the point, The A.M. of the y-coodinates of P and Q  i.e., 2, , , of intersection of tangents at P and Q on the parabola., , , , The orthocentre of the triangle formed by three tangents to the parabola lies on the directrix., , 11. EQUATIONS OF NORMAL IN DIFFERENT FORMS, (i) Point form, The equation of the normal to the parabola y2 = 4ax at a point (x1, y1) is, y1, ( x  x1 ) ., y – y1 = –, 2a, (ii) Parametric form, The equation of the normal to the parabola y2 = 4ax at the point (at2, 2at) is, y + tx = 2at + at3., (iii) Slope form, The eqaution of normal to the parabola y2 = 4ax in terms of slope 'm' is, y = mx – 2am – am 3, Note : The coordinates of the point of contact are (am2 – 2am)., , Ex.7, , Find the equation of a normal at the parabola y2 = 4x which passes through (3, 0)., , Sol., , Equation of normal y = mx – 2am – am3, Here a = 1 and it passes through (3, 0), , , for, , 0 = 3m – 2m – m3, m3 – m = 0, m = 0, ± 1, m=0  y=0, m=1  y=x–3, m = –1  y = – x + 3, 95

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PARABOLA, , Number of tangents drawn from a point to a parabola, Two tangents can be drawn from a point to a parabola. The two tangents are real and distinct or coincident, or imaginary according as the given point lies outside, on or inside the parabola., , 13. EQUATION OF THE PAIR OF TANGENTS, The equation of the pair of tangents drawn from a point P(x1, y1) to the parabola y2 = 4ax is SS1 = T2., Q, P, (x 1, y 1), R, , where S  y2 – 4ax, S1  y12 – 4ax1 and T  yy1 – 2a (x + x1), Locus of point of intersection, The locus of point of intersection of tangent to the parabola y2 = 4ax which are having an angle  between, them given by y2 – 4ax = (a + x)2 tan2 , Note :, , , If  = 0° or  then locus is (y2 – 4ax) = 0 which is the given parabola., , , , If  = 90°, then locus is x + a = 0 which is the directrix of the parabola., , 14. CHORD OF CONTACT, The equation of chord of contact of tangents drawn from a point P(x1, y1) to the parabola y2 = 4ax is, T = 0 where T  yy1 – 2a (x + x1)., Q, Chord, of, contact, , P, (x 1, y 1), R, , Note :, , , The chord of contact joining the point of contact of two perpendicular tangents always passes through, focus., 1, ( y 12  4ax 1 )( y 12  4a 2 ), a, , , , Lengths of the chord of contact is, , , , Area of triangle formed by tangents drawn from (x1, y1) and their chord of contact is, , 1 2, ( y1  4ax 1 )3 / 2 ., 2a, , Ex.10, , Find the area of triangle made by the chord of contact and tangent drawn from point (4, 6) to the parabola, y2 = 8x., , Sol., , Area, Here, , 1, (y 2 – 4ax1)3/2, 2a 1, a = 2, (x1, y1) = (4, 6), , =, ,  Area of triangle , , 1, ( 4)3 / 2, (36  32)3 / 2 , 2, 2.2, 4, , 15. CHORD WITH A GIVEN MID POINT, P, , The equation of the chord of the parabola y2 = 4ax with P(x1, y1) as its middle point is given, by, R(x1 , y1), , T = S1, where T  yy1 – 2a (x + x1) and S1  y12 – 4ax., , Q, , 98

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PARABOLA, , 16. POLE AND POLAR, Let P be a given point. Let a line through P intersect the parabola at two points A and B. Let the tangents, at A anf B intersect at Q. The locus of point Q is a straight line called the polar of point P with respect to, the parabola and the point P is called the pole of the polar., B, Q, A, P(x1, y1), , Equation of Polar of a Point, The polar of a point P(x1, y1) with respect to the parabola y2 = 4ax is T = 0 where T  yy1 – 2a (x + x1)., Coordinate of pole, ,  n 2am , , The pole of the line x + my + n = 0 with respect to the parabola y2 = 4ax is  ,,  , , Conjugate points and conjugate lines, , (i) If the polar of P(x1, y1) passes through Q(x2, y2) then the polar of Q will pass through P and such points, are said to be conjugate points., So two points (x1, y1) and (x2, y2) are conjugate points with respect to parabola y2 = 4ax if yy1 = 2a(x1+x2), (ii) If the pole of a line ax + by + c = 0 lies on the another line a1x + b1y + c1 = 0 then the pole of the second, line will lie on the first and such line are said to be conjugate lines., So two lines 1x + m 1y + n1 = 0 and 2x + m 2y + n2 = 0 are conjugate lines with respect to parabola, y2 = 4ax if, (1n2 + 2n1) = 2 am 1m 2, Note,  The polar of focus is directrix and pole of directrix is focus.,  The polars of all points on directrix always pass through a fixed point and this fixed point is focus.,  The pole of a focal chord lies on directrix and locus of poles of focal chord is a directrix.,  The chord of contact and polar of any point on the directrix always passes through focus., Ex.11, , Find the locus of pole of the focal chords of a parabola., , Sol., , Let the pole is (x1, y1) then equation of polar is yy1 = 2a(x + x1), This passes through the focus (a, 0), , , 0 = 2a (a + x1), ,  x1 = –a, , , , locus of pole is x = –a which is the directrix of parabola., , 17. DIAMETER OF A PARABOLA, Diameter of a parabola is the locus of middle points of a series of its parallel chords., The equation of the diameter bisecting chords of slope m of the parabola y2 = 4ax is y =, Y, , O, , 2a, ., m, , y = 2a/m, X, , 99

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PARABOLA, , Ex.12, , 2, x is y + 2x + 1 = 0 then find its diameter.., The equation of system of parallel chords of the parabola y2 =, 3, , Sol., , Here 4a =, , 2, 3, ,  a=, , 1, and m = –2, 6, , 2a, m,  y = –1/6. This is the required equation of diameter., Diameter is y =, , 18. PREPOSITIONS ON THE PARABOLA,  The tangent at any point P on the parabola bisects the angle between the focal chord through P and the, perpendicular from P on the directrix.,  The portion of a tangent to a parabola cut off between the directrix and the curve subtends a right angle at, the focus.,  Tangents at the extremities of any focal chord intersect at right angles on the directrix.,  Any tangent to a parabola and the perpendicualr on it from the focus meet on the tangents at the vertex.,  The sub tangent at any point on the parabola is twice the abscissa or proportional to sqaure of the ordinate, of the point.,  The sub normal is constant for all points on the parabola and is equal to its semi latus rectum 2a.,  if the tangent and normal at any point P of parabola meet the axes in T and G respectively, then, ST = SG = SP,  If a circle intersect a parabola in four points, then the sum of their ordinates is zero.,  The semi latus rectum of a parabola is the H.M. between the segments of any focal chord of a parabola, i.e. if PQR is a focal chord then 2a =, , 2PQ.QR, ., PQ  QR, ,  The abscissa of point of intersection R of tangents at P(x1, y1) and Q(x2, y2) on the parabola is G.M. of abscissa, x1  x 2 , , of P and Q and ordinate of R is A.M. of ordinate of P and Q thus R  x1x 2 ,, ., 2 , , ,  If vertex and focus of a parabola are on the x-axis at distance a and a' from origin respectively then equation, of parabola y2 = 4(a' – a) (x – a).,  The area of triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents, at these points.,  The area of triangle formed inside the parabola y2 = 4ax is, , 1, (y1 – y2) (y2 – y3) (y3 – y1) when y1,y2,y3, 8a, , are ordinate of vertices of the triangle., 100

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PARABOLA, , Ex.6, Sol., , Ex.7, , Sol., , If the normal to the parabola y2 = 4ax drawn at (a, 2a) meets the parabola again at the point (at2, 2at) then t is, equal to[1] 1, [2] 3, [3] –1, [4] –3, If t’ be the parameter of the given point, then, 2at’ = 2a  t’ = 1, 2, 2, Now, t = –t’ –,  t = –1 –, = –3., Ans.(4), t', 1, The point of intersection of two tangents drawn at the points where the line 7y – 4x = 10 meets the parabola, y2 = 4x, is7 5, 5 7,  5 7,  7 5, [2]  , , [3]   , , [4]   , , [1]  , , 2, 2, 2, 2, 2, 2, , , , , , ,  2 2, 2, If (x1, y1) is the required point; then the given line will be the chord of contact of y = 4x with respect to this point, and its equation will be yy1 = 2(x + x1)., Comparing it with ty – 4x = 10, we get, y1 2 2x1,  , 7, 4 10, , x1 = 5/2 y1 = 7/2, 5 7,  requied point is  ,  ., 2 2, , Ex.8, , A normal is drawn to the parabola y2 = 4ax at the point (2a, –2 2 a) then the length of the normal chord, is[2] 6 2 a, , [1] 4 2 a, Sol., , Sol., , [3] 4 3 a, , Here comparing (2a, –2 2 a) with (am2, –2am) we get m =, Now length of normal chord, =, , Ex.9, , Ans.(2), , 4a, m2, , (1  m 2 )3 / 2 =, , [4] 6 3 a, , 2, , 4a, (1  2)3 / 2 = 2a 3 3  6 3 a, 2, , Ans.[4], , If the tangents at P and Q on a parabola (whose focus is S) meet in the point T, then SP, ST and SQ are in, [1] H.P., [2] G.P., [3] A.P., [4] none of these, Let P(at12, 2at1) and Q (at22, 2at2) be any two points on the parabola y2 = 4ax, then point of intersection of, tangents at P and Q will be, T  [at1t2,a(t1 + t2)], Now, SP = a(t12 + 1), SQ = a(t22 + 1), ST = a ( t 12  1)( t 22  1), ST2 = SP.SQ, , SP, ST and SQ are in G.P., , , , Ans.[2], , Ex.10 If a normal chord of the parabola y2 = 4ax subtend a right angle at the vertex, its slope is, Sol., , [3] ± 3, [1] ± 1, [2] ± 2, If P(at12, 2at1) be one end of the normal, the other say Q(at22, 2at2), , 2, t2 = –t1 – t, 1, , then, Again slope of OP =, , 2, Slope of OQ = t, 2, , From (1) and (2), , 2at 1, at 12, , , , [4] none of these, , .....(1), , 2, t1, , 2 2, ,  1, t1 t 2, t1t2 = –4, , , , , , 4, 2,  t1 , t1, t1, , .....(2), , , , 2,  t1, t1, , t12 = 2  t1 = ±, , 2 Ans.(2), 102

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PARABOLA, 2, , Ex.11 The locus of the point of intersection of perpendicular tangents to the parabola x – 8x + 2y + 2 = 0 is, [1] 2y – 15 = 0, [2] 2y + 15 = 0, [3] 2x + 9 = 0, [4] none of these, Sol., We know that the locus of the perpendicualr tangents of a parabola is its directrix. Now given equation can be, written as, (x – 4)2 = –2(y – 7), Here a = –1/2, so the required locus i.e., the directrix is, y – 7 = –(–1/2), , 2y – 15 = 0, Ans.(1), Ex.12, , The slope of tangents drawn from a point (4, 10) to the parabola y2 = 9x are, , Sol., , 1 1, 1 3, 1 9, ,, [2] ,, [3] ,, 4 3, 4 4, 4 4, The equntion of a tangent of slope m to the parabola y2 = 9x is, , [1], , y = mx +, , [4] None of these, , 9, 4m, , If it passes through (4, 10), then, 10 = 4m +,  (4m – 1) (4m – 9) = 0  m =, , 9,  16m2 – 40m + 9 = 0, 4m, 1 9, ,, 4 4, , Ans.(2), , Ex.13 Tangents are drawn from the point (–2, –1) to the parabola y2 = 4x. If  is the angle between these tangents then, tan  equal[1] 3, [2] 1/3, [4] 2, [4] 1/2, 2, Sol., Any tangent to y = 4x is, y = mx + 1/m., If it is drawn from (–2, –1), then, –1 = –2m + 1/m, , 2m2 – m – 1 = 0, If m = m1, m2 then m1 + m2 = 1/2,, m1m2 = –1/2, , , tan  =, , , , m1  m2, , 1  m1m 2, , (m1  m2 )2  4m1m 2, 1  m1m 2, , 1/ 4  2, 3, 1  1/ 2, , Ex.14 Which of the following lines, is a normal to the parabola y2 = 16x, [1] y = x – 11 cos – 3 cos 3, [2] y = x – 11 cos – cos 3, [3] y = (x – 11) cos + cos 3, [4] y = (x – 11) cos – cos 3, Sol., Here, a=4, Condition of normalily c = –2am – am3, (1) and (2) are not clearly the answer as, m=1, for (3), (4) m = cos , c = –2(4) cos  – 4 cos3, = –8 cos – (3 cos + cos 3), = – 11 cos – cos 3, , Ans.(1), , Ans.(4), 103

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PARABOLA, , EXERCISE # 1, Q.1, , If focus of the parabola is (3, 0) and length of latus rectum is 8, then its vertex is, [1] (2, 0), [2] (1, 0), [3] (0, 0), [4] (– 1, 0), , Q.2, , The area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus, rectum is, [1] 16 sq. units, [2] 12 sq. units, [3] 18 sq. units, [4] 24 sq. units, , Q.3, , Any point on the parabola whose focus is (0, 1) and the directrix is x + 2 = 0 is given by, [1] (t2 + 1, 2t + 1), [2] (t2 + 1, 2t – 1), [3] (t2, 2t), [4] (t2 – 1, 2t + 1), , Q.4, , If (0, a) be the vertex and (0, 0) be the focus of a parabola then its equation will be, [1] y2 = 4a (a + x), [2] x2 = 4a (a – y), [3] x2 = 4a (a + y), [4] y2 = 4a (a – x), , Q.5, , If vertex and focus of a parabola are on x-axis and at distance p and q respectively from the origin then its, equation is, [1] y2 = – 4 (p – q) (x + p), [2] y2 = 4 (p – q) (x – p), [3] y2 = – 4 (p – q) (x – p), [4] none of these, , Q.6, , The distance between the focus and the directrix of the parabola x2 = – 8y, is, [1] 8, [2] 2, [3] 4, [4] 6, , Q.7, , If (2, 0) and (5, 0) are the vertex and focus of a parabola respectively then its equation is, [1] y2 = – 12x – 24, [2] y2 = 12x – 24, [3] y2 = 12x + 24, [4] y2 = – 12x + 24, , Q.8, , The point of intersection of the latus rectum and axes of the parabola y2 + 4x + 2y – 8 = 0 is, [1] (5/4, – 1), [2] (7/5, 5/2), [3] (9/4, – 1), [4] None of these, , Q.9, , The circle drawn with any focal chord of a parabola as diameter touches, [1] tangents at the vertex, [2] directrix, [3] axis of parabola, [4] latus - rectum, , Q.10, , Which of the following are not parametric coordinates of any point on the parabola y2 = 4ax, [1] (a/m2, 2/m), [2] (a, 2a), [3] (at2, 2at), [4] (am2, – 2am), , Q.11, , The parametric equations of the parabola y2 – 12x – 2y – 11 = 0 are, [1] x = 3 t2 – 1, y = 6t + 1, [2] x = 3t2 + 1, y = 6t – 1, [3] x = 6t + 1, 3t2 – 1, [4] none of these, , Q.12, , x – 2 = t2, y = 2t are the parametric equations of the parabola, [1] y2 = – 4x, [2] y2 = 4x, [3] x2 = – 4y, , [4] y2 = 4 (x – 2), , Q.13, , Lines y = x and y = – x intersect the parabola y2 = 4x at A and B other than the origin. The length AB is, [1] 12, [2] 8, [3] 4, [4] 16, , Q.14, , The length of the intercept made by the parabola 2y2 + 6y = 8 – 5x on y-axis is, [1] 7, [2] 5, [3] 3, [4] 1, 104

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PARABOLA, , Q.15, , Length of the chord intercepted by the parabola y = x2 + 3x on the line x + y = 5 is, [2], , [1] 6 2, , [3] 6 3, , 2, , [4] none of these, , Q.16, , If length of the two segments of focal chord to the parabola y2 = 8ax are 2 and 4, then the value of a is, [1] 1/3, [2] 2/3, [3] 4/3, [4] 4, , Q.17, , The normal at (a, 2a) on y2 = 4ax, meets the curve again at (at2, 2at), then the value of t is, [1] 3, [2] 1, [3] – 1, [4] – 3, , Q.18, , If (2, – 8) is at an end of a focal chord of the parabola y2 = 32x then the other end of the chord is, [1] (32, 32), [2] (– 2, 8), [3] (32, – 32), [4] none of these, , Q.19, , The length of the chord of the parabola y2 = 4x which passes through the vertex and makes 30º angle with x-axis, is, [1], , Q.20, , [2] 3/2, , 3 /2, , [2] 3 5, , bc, 2, , [2], , [3] 8, , [4] none of these, , bc, bc, , [3], , 2bc, bc, , [4], , bc, , The mid point of the chord 2x + y – 4 = 0 of the parabola y2 = 4x is, , 5, 2, , , , , [1]  , 1, , Q.23, , 3, , If b and c are the lengths of the segments of any focal chord of a parabola y2 = 4ax, then the length of the semilatus rectum is, [1], , Q.22, , [4], , The length of chord of contact of tangents drawn from point (– 2, – 1) to the parabola y2 = 4x is, [1] 2 2, , Q.21, , [3] 8 3, , , , , [2]  1,, , 5, 2 , , 3, , , 1, 2, , , [3] , , , , , [4]  1, , , 3, 2 , , If (at2, 2at) are the coordinates of one end of a focal chord of the parabola y2 = 4ax, then the coordinate of the, other end are [1] (at2, – 2at), , [2] (– at2 , – 2at), ,  a 2a , ,, 2, t ,  t, , [3] , ,  a 2a , , , 2, t ,  t, , [4] , , Q.24, , The diameter of the parabola y2 = 6x corresponding to the system of parallel chords 3x – y + C = 0 is, [1] y – 1 = 0, [2] y – 2 = 0, [3] y + 1 = 0, [4] y + 2 = 0, , Q.25, , The point of contact of the line 2x – y + 2 = 0 with the parabola y2 = 16 x is, [1] (3, 4), [2] (2, 4), [3] (1, 4), [4] (– 2, 1), , Q.26, , The area of the triangle formed by tangents drawn from the point (x1, y1) to the parabola y2 = 8x and its chord of, contact is, [1] (y12 – 8x1)3/2, , [2], , 1 2, (y – 8x1)3/2, 4 1, , [3], , 1, (y 2 – 8x1)3/2, 2 1, , [4], , 1, (y 2 – 8x1)1/2, 4 1, 105

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PARABOLA, , Q.27, , The equation of the chord of contact of tangents drawn from the point (2, 3) to the parabola y2 + x = 0 is, [1] 6y – x = 2, [2] 3y + x = 2, [3] 6y + x + 2 = 0, [4] 3y – x = 2, , Q.28, , The equation of a system of parallel chrods to parabola y2 =, diameter is, [1] 25x = 56, , [2] 56x = 25, , 25x, is 4x – y +  = 0 then equation to corresponding, 7, , [3] 25 y = 56, , [4] 56y = 25, , Q.29, , The equation of the tangent at vertex to the parabola 4y2 + 6x = 8y + 7 is, [1] x = 11/6, [2] y = 2, [3] x = – 11/6, [4] y = – 2, , Q.30, , The equation of tangent to the parabola x2 = y at one extremity of latus rectum in the first quadrant is, [1] y = 4x + 1, [2] x = 4y + 1, [3] 4x + 4y = 1, [4] 4x – 4y = 1, , Q.31, , The line x + my + n = 0 will touch the parabola y2 = 4ax if, [1] m = an2, [2] mn = a2, [3] n = am2, , [4] mn = a, , Q.32, , The locus of the point of intersection of perpendicular tangents to the parabola y2 – 6y + 24x – 63 = 0 is, [1] 2y – 9 = 0, [2] x – 9 = 0, [3] x – 6 = 0, [4] none of these, , Q.33, , If a tangent to the parabola 4y2 = x makes an angle of 60º with the x-axis, then its point of contant is, ,  1, 1 , ,, ,  48 8 3 , , [1] , Q.34, ,  3, , 3, , [2]  16 , 8 , , , ,  1, 1 , ,, ,  48 8 3 , , [3] , ,  3, , Area of triangle formed by the tangents at three points t1, t2 and t3 of the parabola y2 = 4ax is, [1], , a, (t – t )(t – t )(t – t ), 2 1 2 2 3 3 1, , [2] a2(t1 – t2) (t2 – t3)(t3 – t1), , [3], , a2, (t + t )(t + t ) (t + t ), 2 1 2 2 3 3 1, , [4], , a2, (t – t ) (t – t )(t – t ), 2 1 2 2 3 3 1, , Q.35, , From the point (2, – 2), to the parabola 3x2 + 8y = 0, we can draw, [1] One tangent and one normal, [2] Two tangents, [3] One normal, [4] Three normals, , Q.36, , If the line x + y = k is a normal to the parabola y2 = 4x, then the value of k will be, [1] 2, [2] 1, [3] 3, [4] 4, , Q.37, , The slope of normal to the parabola x2 + 4y = 0 at the point (2, – 1) is, [1] – 1, [2] 1, [3] 3/2, , [4] – 3/2, , Find the equation of normal to the curve 2y = 3 – x2 at the point (1, 1), [1] x + y + 1 = 0, [2] x + y = 0, [3] x – y + 1 = 0, , [4] x – y = 0, , Q.38, , Q.39, , 3, , [4]  16 ,  8 , , , , The equation of the normal to the parabola x2 = 8y whose slope is 1/m is, [1] y = mx – 2m – m3 [2] x = my – 4m – 2m3 [3] x = my – 2m – m3, [4] 4y = mx + 4m + 2m3, 106

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PARABOLA, , Q.40, , The equation of the normal to the parabola y2 + 12x = 0 at the upper end of its latus rectum is, [1] x – y – 8 = 0, [2] x + y + 9 = 0, [3] x – y – 9 = 0, [4] x – y + 9 = 0, , Q.41, , The equation of the normal having slope m of the parabola y2 = x + a is, [1] y = mx – am – am3 [2] y = mx – 2am – am3 [3] 4y = 4mx + 4am – 2m – m3, , [4] 4y = 4mx + 2am – am3, , Q.42, , If normals drawn at two points of parabola y2 = 4ax meet on the parabola then the product of the ordinates of, these points is, [1] 4a, [2] 8a2, [3] 8a, [4] 4a2, , Q.43, , The equation of the normal at the ends of the latus rectum of the parabola y2 = 4ax are given by, [1] x2 – y2 – 6ax + 9a2 = 0, [2] x2 – y2 – 6ay + 9a2 = 0, [3] x2 – y2 – 6ax – 6ay + 9a2 = 0, [4] none of these, , Q.44, , The line x + my + n = 0 is a normal to the parabola y2 = 4ax if, [1] a (2 + 2m2) + m2n = 0, [2] a (2 2 + m2) = – m2n, [3] a (2 + 2m2) = m2n, [4] a (22 + m2) = 2m2n, , Q.45, , PQ is double ordinate of y2 = 4ax. The locus of its point of trisection is, [1] y2 = 2ax, [2] 3y2 = 4ax, [3] 9y2 = 4ax, , [4] 9y2 = 2ax, , Q.46, , The length of the subnormal at any point of the parabola y2 = 8x is, [1] 4, [2] 2 (abscissa of the point), [3] 8, [4] None of these, , Q.47, , The extremities of the latus rectum of the parabola x2 + 8y = 0 are, [1] (–4, –2); (4, 2), , Q.48, , [2] (4, –2); (–4, 2), , The equations x =, , [2] y + 2 = 0, , [4] y – 2 = 0, , [2] a circle, , [3] a parabola, , [4] none of these, , The equation of the parabola whose vertex and focus are (1, 1) and (3, 1) respectively, will be, [1] (y – 1)2 = 8(x – 3), , Q.51, , [3] x – 2 = 0, , t, t2, represent, ,y=, 4, 4, , [1] a straight line, Q.50, , [4] (4, 2); (–4, 2), , The equation of the axis of the parabola x2 – 4x – 3y + 10 = 0 is, [1] x + 2 = 0, , Q.49, , [3] (–4, –2); (4, –2), , [2] (x – 1)2 = 8(y – 1), , [3] (x – 3)2 = 8(y – 1), , [4] (y – 1)2 = 8(x – 1), , The equation of the parabola passing through the point (6, –3) whose vertex is the origin and axis is y-axis, will, be, [1] y2 = 12x + 6, , Q.52, , [3] x2 = –12y, , [4] y2 = –12x + 6, , If the parabola (y + 1)2 = a(x – 2) passes through the point (1, –2) then equation of its directrix is, [1] 4x + 9 = 0, , Q.53, , [2] x2 = 12y, , [2] 4x – 9 = 0, , [3] 4y + 3 = 0, , [4] 4y – 3 = 0, , The length of the focal chord of the parabola y2 = 4ax at point ‘t is, [1] a(t + 1/t)2, , [2] a(1 + t2), , [3] a(t + 1/t), , [4] a(1 + 1/t)2, , 107

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PARABOLA, , Q.54, , 2, , The equation of the tangent to the parabola x = 8y perpendicular to the line x = 2y – 1, is, [1] x + 2y + 8 = 0, , Q.55, , Q.58, , [2] m1m2 = 1, , [3] m1m2 = –1, , [4] none of these, , [2] p, , [3] 4p, , [4] none of these, , The line 2x + y – 1 = 0 meets the parabola y2 = 4x at, [1] two real and different points, , [2] two coincident points, , [3] tow imaginary points, , [4] none of these, , Equation of the chord of the parabola y2 = 6x which is bisected at (–1, 1) is, [1] y – 3x = 4, , Q.59, , [4] none of these, , The length of the subnormal at any point of the parabola y2 = 2px is, [1] 2p, , Q.57, , [3] x + y + 8 = 0, , If y + b = m1(x + a) and y + b = m2(x + a) are two tangents of the parabola y2 = 4ax, [1] m1 + m2 = 0, , Q.56, , [2] 2x + y + 8 = 0, , [2] y – 3x + 4 = 0, , [3] 3x – y = 0, , [4] 3x – y = 1, , The length of the chord of the parabola y2 = 4ax which passes through the vertex and makes  angle with x-axis,, is, [1] 4a sin  sec2 , , Q.60, , [2] 4a cos  cosec2 , , [4] 4a sin  cos2 , , The area of the triangle with vertices t1, t2 and t3 on the parabola y2 = 4ax is, [1] a2(t1 – t2)(t2 – t3)(t3 – t1), , a2, (t – t )(t – t )(t – t ), 2 1 2 2 3 3 1, [4] none of these, , [2], , [3] 2a2 (t1 – t2)(t2 – t3)(t3 – t1), , Q.61, , [3] 4a cos  sin2 , , Tangents are drawn from the point (–1, 2) to the parabola y2 = 4x. The area of the triangle formed by these, tangents and their chord of contant is, [1] 8, , [2] 8 2, , [3] 8 3, , [4] none of these, , 108

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PARABOLA, , EXERCISE # 2, Q.1, , The coordinates of a point on the parabola y2 = 8x whose focal distance is 4 is, [1] (2, 4), [2] (4, 2), [3] (2, – 9), [4] (4, – 2), , Q.2, , If focus and vertex of a parabola are respectively (1, –1) and (2, 1), then its directrix is, [1] x + 2y – 9 = 0, [2] 2x – y – 9 = 0, [3] x + 2y + 9 = 0, [4] none of these, , Q.3, , Two tangents are drawn from point (– 2, – 1) to the parabola y2 = 4x, if  is the angle between these tangents,, then tan  equals, [1] 3, [2] 1/3, [3] 2, [4] 1/2, , Q.4, , If a tangent line at a point P on a parabola makes angle  with its focal distance, then angle between the tangent, and axis of the parabola is, [1] , [2] /2, [3] 2, [4] 90º, , Q.5, , The equation of the common tangents to the parabolas y2 = 4x and x2 = 32y is, [1] x + 2y = 4, [2] x = 2y + 4, [3] x = 2y – 4, [4] x + 2y + 4 = 0, , Q.6, , Two tangents drawn on parabola y2 = 4ax are making angle 1 and 2 with x-axis and if 1 + 2 = 2 then locus, of the point of intersection of these tangent is, [1] x = (y – a) tan 2, [2] y = (x – a) cos 2, [3] y = (x – a) tan 2, [4] y = (x + a) tan 2, , Q.7, , A tangent to the parabola y2 = 4ax at P(p, q) is perpendicular to the tangent at the other point Q, then coordinates, of Q are, [1] (a2/p, – 4a2/q), [2] (–a2/p, – 4a2/q), [3] (– a2/p, 4a2/q), [4] (a2/p, 4a2/q), , Q.8, , At which point the normal drawn at upper end of latus rectum of the parabola y2 = – 12x meets its axis, [1] (0, – 9), [2] (– 9, 0), [3] (9, 0), [4] none of these, , Q.9, , The equation of normal to the parabola y2 = 4x passing through the point (3, 0) is, [1] y + x = 3, [2] y = x – 3, [3] y = 0, [4] all three correct, , Q.10, , Three normals to the parabola y2 = x are drawn through a point (C, 0), then, [1] C = 1/4, [2] C = 1/2, [3] C > 1/2, [4] none of these, , Q.11, , The locus of point of intersection of two perpendicular normals drawn on the parabola y2 = 4ax is, [1] y2 = a (x – 3a), [2] y2 = a (x + 2a), [3] y2 = a (x – 2a), [4] y2 = a (x + 3a), , Q.12, , The coordinate of the point where the normal to the parabola drawn at its point (2, 4) meets it again are, [1] (18, – 12), [2] (– 18, 12), [3] (18, 12), [4] (– 18, – 12), , Q.13, , The locus of mid points of chord of the parabola x2 + 4y = 0 passing through its focus is, [1] x2 + 2y + 2 = 0, [2] y2 + 2x + 2 = 0, [3] x2 + 2y = 0, [4] none of these, , Q.14, , If the chord y = mx + c subtends a right angle at the vertex of the parabola y2 = 4ax, then the value of c is, [1] – 4am, [2] 4am, [3] – 2am, [4] 2 am, , 109

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PARABOLA, , 17, and is in first quadrant, then equatin of normal at this, 4, , Q.15, , If the focal distance of a point on the parabola y2 = x is, , Q.16, , point is, [1] y – 4x = 18, [2] 4y – x = 4, [3] 4x + y = 18, [4] 4x + y + 18 = 0, 2, The angle between tangents to the parabola y = 4a(x – a) drawn from the origin is, [1] 90°, , Q.17, , [4] 45°, , The slope of tangent lines drawn from (3, 8) to the parabola y2 = –12 x are[1] 3, 1/3, , Q.18, , [3] tan–1 (1/2), , [2] 30°, , [2] –3, –1/3, , [3] 3, –1/3, , [4] –3, 1/3, , Two tangents of the parabola y2 = 8x meet its tangent at vertex at points P and Q. If PQ = 4, then the locus of the, point of intersection of these two tangents is, [1] y2 = 8(x – 2), , Q.19, , [2] y2 = 8(x + 2), , [3] x2 = 8(y – 2), , [4] x2 = 8(y + 2), , The set of points on the axis of the parabola y2 – 2y – 4x + 5 = 0 from which all the three normals to the parabola, are real is[1] {(x , 1) : x  3}, , Q.20, , [2] greater than a, , 10, , Q.26, , [2] PQ = 80, , 10, , [3] PQ =, , 10, , [4] 9PQ =, , 10, , [2] cy2 – x2 + 2ax = 0, , 2, , [3] cx2 + y2 – 2ax = 0, , [4] cx2 – y2 + 2ax = 0, , The locus of point of intersection of tangents inclined at angle 45° to the parabola y2 = 4x is[2] y2 – 4x = x2, , [3] y2 – 4x = (x + 2)2, , [4] none of these, , If a focal chord of a parabola y2 = 4ax makes an angle  with its axis then the length of perpendicular from vertex, to this chord is[1] a tan , , Q.25, , [4] none of these, , Two tangents drawn on parabola y = 4 ax are making angle 1 and 2 with x-axis and if tan 1 + tan22 = c, then, locus of the point of intersection of these tangent is-, , [1] y2 – 4x = (x + 1)2, Q.24, , [3] greater than 2a, , 2, , [1] y2 = cx2, Q.23, , [4] {(x, –3) : x  3}, , If the normal at P (18, 12) to the parabola y2 = 8x cuts it again at Q then[1] 9PQ = 80, , Q.22, , [3] {(x, 3) : x  1}, , From a point on x-axis, if three real normals are drawn to the parabola y2 = 4 ax (a > 0), then abscissa of the point, must be[1] less than a, , Q.21, , [2] {(x ,–1) : x  1}, , [2] a cos , , [3] a sin , , [4] a sec , , The equation of the common tangent to the parabolas y2 = 4ax and x2 = 4by is, [1] xa1/3 + yb1/3 + (ab)2/3 = 0, , [2] xa1/3 + yb1/3 = (ab)2/3, , [3] xa2/3 + yb2/3 + (ab)2/3 = 0, , [4] xa2/3 + yb2/3 = (ab)2/3, , From vertex O of the parabola y2 = 4ax perpendicular is drawn at a tangent to the parabola. If it meets the tangent, and the parabola at point P and Q respectively then OP.OQ is equal to[1] constant, , Q.27, , [2] 1, , [3] –1, , [4] 2, , The angle made by a double ordinate of length 8a at the vertex of the parabola y2 = 4ax is[1], , , 3, , [2], , , 2, , [3], , , 4, , [4], , , 6, , 110

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PARABOLA, , EXERCISE # 3, Q.1, , The locus of the middle point of the chords of parabola y2 = 4ax, which passes through the origin is-[RPET-97], [1] y2 = 2ax, , Q.2, , ab, (b  a), , [2] y = x + 2, , [2] m = an2, , [2] y = 3/8, , [2], , b, (b  a), ,  n  2am , , [2]  ,,  , , , [2] 1/2, , 2, , [3] y = x – 2, , [RPET-97, AIEEE-2002], [4] y = –x + 2, , [3] mn = a2, , [RPET-98], [4] none of these, , [3] y = –9/8, , [RPET-98], [4] none of these, , [3], , a, (b  a), , [4], ,   n 2am , ,, , [3] ,  ,  , , [3] 1, , [RPET-98], , ab, (a  b ), [RPET-98], , [4] none of these, [RPET-98], [4] 4, , [2], , 3, , [3] –1, , [4] 1, , The polar of the focus of parabola will be[1] Directrix, , Q.13, ,  1 1, [4]   , ,  2 2, , If the normal chord of a parabola y2 = 4ax subtends an angle of /2 at vertex then slope of normal is-[RPET-99], [1], , Q.12, , 1 5, [3]  , , 2 8, , The length of LR of the parabola 4x2 – 4x – 2y + 3 = 0 is[1] 2, , Q.11, ,  5 1, [2]   , ,  8 2, , [RPET-97], , The pole of the line x + my + n = 0 w.r.t. the parabola y2 = 4ax will be n 2am , , [1]  ,,  , , , Q.10, , [4] (–2a, a), (2a, a), , The intercept of the focal chord of parabola y2 = 4ax are b and k, then the value of k will be-, , [1], Q.9, , [3] (a, –2a), (2a, a), , [RPET-97], , The equation of directrix of the parabola 4x2 – 4x – 2y + 3 = 0 will be[1] y = 7/8, , Q.8, , [2] (–a, 2a), (2a, a), , [RPET-97], , [4] 24, , If the line x + my + n = 0 touches the parabola y2 = 4ax then[1] n = am2, , Q.7, , [3] –24, , The common tangent of the circle x2 + y2 = 2 and parabola y2 = 8x will be[1] y = x + 1, , Q.6, , [2] –8, , The coordinate of focus of the parabola 4y2 – 6x – 4y = 5 is-, ,  8 , [1]   , 2 ,  5 , Q.5, , [4] x2 = 4ay, , The extremities of the latus rectum of the parabola x2 = 4ay are[1] (2, 2a) (2a, –a), , Q.4, , [3] y2 = 4ax, , If the line 2x + y +  = 0 be the normal of the parabola y2 = –8x then the value of  will be[1] –16, , Q.3, , [2] y2 = ax, , [2] Axis, , [RPET-99], [3] Latus rectum, , [4] none of these, , If the line x + my + n = 0 touches the parabola x2 = y then[1] 2 = 4nm, , [2] 2 = 2nm, , [3]  = 4n2m2, , [RPET-99], [4] none of these, 111

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PARABOLA, , Q.14, , 2, , The tangent drawn from the ends of latus rectum of y = 12x meets[1] Directrix, , Q.15, , [2] Vertex, , [3] Focus, , [RPET-2000], [4] none of these, , The vertex and focus of a parabola on x-axis situated at a and a’ distances form origin respectively then its, equation will be[1] y2 = 4(a – a’) (x – a), , Q.16, ,  a 2a , , [4]  2 ,, m m , , [2] (0, 2a) & y = –2a, , [RPET-2001], [3] (0, –2a) & y = 2a, , [4] (2a, 0) & x = –2a, , [2] ST  SG = SP, , [3] ST = SG  SP, , [4] ST = SG = SP, , [2] 2, , [3] 3, , [RPET-2001], [4] 4, , [2] (at2 – 2at), , [RPET-2002], [3] (–2at, at2), , [4] (2at, at2), , The point of intersection of tangents drawn to the parabola y2 = 4ax at (at12, 2at1) (at22, 2at2) is- [RPET-2002], [1] (at1t2, a(t1 + t2)), , Q.23, ,   a  2a , , [3]  2 ,, m , m, , [RPET-2001], , The parametric coordinates of x2 = 4ay, will be[1] (at2, 2at), , Q.22, , [4] 1, , If line y = mx + 2 is tangent to the parabola y2 = 16x then m is[1] 1, , Q.21, ,  a 2a , , [2]   2 ,,  m m, , [2] (a (t1 + t2), a t1t2), , [3] (t12 t22, t1 + t2), , [4] none of these, , A tangent drawn at point P to the parabola meets the directrix at point Q, if S is focus of parabola then angle PSQ, would be[1] /4, , Q.24, , [RPET-2000], , If the tangent and normal at any point P of parabola meet the axes in T and G respectively then- [RPET-2001], [1] ST = SG . SP, , Q.20, , [3] 2, , Focus & directrix of x2 = –8ay is[1] (–2a, 0) & x = 4, , Q.19, , [2] 3 3, , At what point line y = mx + c touches the parabola y2 = 4ax is a  2a , , [1]  2 ,, m , m, , Q.18, , [2] y2 = –4(a – a’) (x + a) [3] y2 = –4(a – a’) (x – a) [4] none of these, , The length of the normal of the parabola y2 = 4x which subtend right angle at the vertex is[1] 6 3, , Q.17, , [RPET-2000], , [RPET-2002], [2] /3, , [3] /2, , [4] , , The normal drawn at a point (at12, 2at1) of the parabola y2 = 4ax meets it again in the point (at22, 2at2), then[RPET-2003], [1] t1 = 2t2, , Q.25, , [3] t1t2 = –1, , [4] none of these, , If a tangent of y2 = ax made angle of 45° with the x-axis then its point of contact will be[1] (a/2, a/4), , Q.26, , [2] t12 = 2t2, , [2] (–a/2, a/4), , [3] (a/4, a/2), , [RPET-2003], , [4] (–a/4, a/2), , The equation of the line which is parallel to the directrix and at 6 unit distance of parabola y2 = 8x is[RPET-2003], [1] x = 4, , Q.27, , [2] x = 8, , [3] x = 0, , [4] none of these, , If line 2x + y + k = 0 is normal to the parabola y2 + 8x = 0 then k is[1] 12, , [2] –12, , [3] 24, , [RPET-2003], [4] –24, 112

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PARABOLA, , Q.28, , If a  0 and the line 2bx + 3cy + 4d = 0 passes through the point of intersection of parabolas y = 4ax and, 2, , x2 = 4ay, then[1] d2 +(2b –3c)2 = 0, Q.29, , [AIEEE-2004], [2] d2 + (3b + 2c)2 = 0, , [3] d2 + (2b + 3c)2 = 0, , [4] d2 + (3b – 2c)2 = 0, , The point on the parabola y2 = 18x at which the ordinate increases at twice the rate of the abscissa is[AIEEE-2004],  9 9, [1]   , ,  8 2, , Q.30, , (2) x2 + 4y + 2 = 0, , (3) y2 + 4x + 2 = 0, , (4) y2 - 4x + 2 = 0, , The point of intersection of the tangents at the ends of the latus rectum of the parabola y2 = 4x is[1] (–1, 0), , Q.32, , 9 9, [4]  , , 8 2, , [3] (2, 4), , Let P be the point (1, 0) and Q a point on the parabola y2 = 8x. The locus of mid point of PQ is [AIEEE 2005], (1) x2 - 4y + 2 = 0, , Q.31, , [2] (2, –4), , [2] (1, 0), , [3] (0, 1), , [4] none of these, , Consider a circle with centre lying on the focus of the parabola y2 = 2px such that it touches the directrix of the, parabola. Then a point of intersection of the circle and the parabola is[1] (p/2, p), , Q.33, , [2] (–p/2, p), , [2] 9, , [3] a parabola, , [IIT-98], [4] a hyperbola, [IIT-99], , [3] –9, , [4] –3, , If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the values of k is[1] 1/8, , Q.36, , [4] none of these, , If x + y = k is normal to y2 = 12x, then k is[1] 3, , Q.35, , [3] (–p/2, –p), , [IIT-95], , The curve described parametrically by x = t2 + t + 1, y = t2 – t + 1 represents[1] a pair of straight lines [2] an ellipse, , Q.34, , [IIT-94], , [2] 8, , [3] 4, , [IIT-2000], , [4] 1/4, , Above x-axis, the equation of the common tangents of the circle (x – 3)2 + y2 = 9 and parabola y2 = 4x is[IIT-2001], [1], , Q.37, , 3 y  3x  1, , 3 y  ( x  3), , [3], , 3 y  x3, , [4], , 3 y  (3 x  1), , The equation of the directrix of the parabola y2 + 4y + 4x + 2 = 0 is-, , [1] x = –1, Q.38, , [2], , [2] x = 1, , [3] x = –, , 3, 2, , [IIT-2001], [4] x , , 3, 2, , The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y2 = 4ax is, another parabola with directrix[1] x = –a, , Q.39, , [2] x = –a/2, , [IIT-2002], [3] x = 0, , [4] x = a/2, , A focal chord of y2 = 16x is a tangent to (x – 6)2 + y2 = 2. Then the possible values of the slope of this chord are[IIT-2003], [1] {–1, 1}, , [2] (–2, 2}, , [3] (–2, 1/2}, , [4] {2, –1/2}, 113

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PARABOLA, , Q.40, , 2, , The angle between tangents drawn from the point (1, 4) on the parabola y = 4x is[1], , , 6, , , 4, , [2], , [3], , , 3, , [4], , [IIT-2004], , , 2, , a3 x 2 a 2 x, ,  2a is, 3, 2, , Q.41, , The locus of the vertices of the family parabola y , , Q.42, , 35, 64, 105, 3, [2] xy =, [3] xy =, [4] xy =, 16, 105, 64, 4, 2, Angle between the tangents to the curve y = x – 5x + 6 at the points (2, 0) and (3, 0) is, , [AIEEE-2006], , [1] xy =, , [1], Q.43, , , 2, , [2], , , 6, , The equation of a tangent to the parabola, , [3], , , 4, , [4], , , 3, , y 2  8 x is y  x  2 . The point on this line from which the other, , tangent to the parabola is perpendicular to the given tangent is, [1] (2, 4), , [AIEEE-2006], , [2] (–2, 0), , [3] (–1, 1), , [AIEEE-2007], [4] (0, 2), , Passage :- Consider the circle x2 + y2 = 9 and the parabola y2 = 8x. They intersect at P and Q in the first and the, fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the, parabola at P and Q intersect the x-axis at S :, Q.44, , The ratio of the areas of the triangles PQS and PQR is :, [1] 1 : 2, , Q.45, , [4] 1 : 8, , [2] 3 3, , [IIT-JEE-2007], [3] 3 2, , [4] 2 3, , The radius of the incircle of PQR is equal to :, [1] 4, , Q.47, , [3] 1 : 4, , The circumradius of PRS is equal to :, [1] 5, , Q.46, , [2] 1 : 2, , [IIT-JEE-2007], , [2] 3, , Statement I : The curve y  , , [IIT-JEE-2007], [3] 8/3, , [4] 2, , x2,  x  1 is symmetric with respect to the line x = 1, because, 2, , Statement II : A parabola is symmetric above its axis. Then :, , [IIT-JEE-2007], , [1] Statement I is true, statement II is true and II is a correct explanation for I., [2] Statement I is true, statement II is true but II is not a correct explanation for I., [3] Statement I is true, statement II is false., [4] Statement I is false, statement II is true., Q.48, , Consider two curves :, , C1 : y2 = 4x, C2 : x2 + y2 – 6x + 1 = 0 Then :, , [IIT-JEE-2008], , [1] C1 and C2 touch each other only at one point, [2] C1 and C2 touch each other exactly at two points, [3] C1 and C2 intersect (but do not touch) at exactly two points, [4] C1 and C2 neither intersect nor touch each other, , 114

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PARABOLA, , Q.49, , A parabola has the origin as its focus and the line x = 2 as the directrix. then the vertex of the parabola is at :, [AIEEE-2008], [1] (1, 0), , Q.50, , [2] (0, 1), , [3] (2, 0), , [4] (0, 2), , Through ther vertex O of the parabola y2 = 4x chords OP and OQ are drawn at right angles to each other. The, locus of the middle point of PQ is :, [1] y2 = x – 8, , Q.51, , [3] y2 = 2x – 8, , [4] y2 = –2x + 8, , If tangents drawn from a point P on the parabola y2 = 4x are perpendicular, the locus of P is :, [1] x = 2, , Q.52, , [2] y2 = x + 8, , [Orissa (JEE) -2008], , [2] x = 1, , [3] x = –1, , [4] x = 1, , Let A and B be two distinct points on the parabola y2 = 4x. If the axis of the parabola touches a circle of radius r, having AB as its diameter, then the slope of the line joining A and B can be :, [1] –1/r, , Q.53, , [2] 1/r, , [3] 2/r, , [IIT-JEE-2010], [4] –2/r, , The shortest distance between line y – x = 1 and curve x = y2 is :, [1], , Q.54, , 3, 4, , [2], , 3 2, 8, , [AIEEE-2011], , 8, [3], , 3 2, , 4, [4], , 2, , Let (x, y) be any point on the parabola y2 = 4x. Let P be the point that divides the line segment from (0, 0), to (x, y) in the ratio 1 : 3. Then the locus of P is :, [1] x2 = y, , Q.55, , [2] y2 = 2x, , [IIT-JEE-2011], [3] y2 = x, , [4] x2 = 2y, , Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by :[IIT-JEE-11], [1] y – x + 3 = 0, , Q.56, , [AIEEE-2010], , [2] y + 3x – 33 = 0, , [3] y + x – 15 = 0, , [4] y – 2x + 12 = 0, , Given : A circle 2 x 2  2 y 2  5 and a parabola y 2  4 5 x ., , [JEE Mains – 2013], , Statement – I :- An equation of a common tangent to these curves is y  x  5 ., , 5, (m  0) is their common tangent, then m satisfies m 4  3m 2  2  0 ., m, (1) Statement – I is true, Statement – II is false, Statement – II :- If the line, y  mx , , (2) Statement – I is false, Statement – II is true, (3) Statement – I is true, Statement – II is true, Statement–II is a correct explanation for Statement–I, (4) Statement–I is true, Statement–II is true, Statement– II is not a correct explanation for Statement – I, Q.57, , The slope of the line touching both the parabolas y 2  4x and x 2  32 y is :, (1), , 1, 2, , (2), , 3, 2, , (3), , 1, 8, , [JEE Mains – 2013], (4), , 2, 3, , 115

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PARABOLA, , ANSWER KEY, EXERCISE - 1, Que. 1, Ans. 2, Que. 26, Ans. 2, Que. 51, Ans. 3, , 2, 3, , 3, 4, , 4, 2, , 5, 3, , 6, 3, , 7, 2, , 8, 1, , 9, 2, , 10, 1, , 11, 1, , 12, 4, , 13, 2, , 14, 2, , 15, 1, , 16, 2, , 17, 4, , 18, 1, , 19, 3, , 20, 2, , 21, 3, , 22, 1, , 23, 4, , 24, 1, , 25, 3, , 27, 3, 52, 2, , 28, 4, 53, 1, , 29, 1, 54, 2, , 30, 4, 55, 3, , 31, 3, 56, 2, , 32, 2, 57, 1, , 33, 1, 58, 1, , 34, 4, 59, 2, , 35, 4, 60, 1, , 36, 3, 61, 2, , 37, 2, , 38, 4, , 39, 2, , 40, 4, , 41, 3, , 42, 2, , 43, 1, , 44, 1, , 45, 3, , 46, 1, , 47, 3, , 48, 3, , 49, 3, , 50, 4, , EXERCISE - 2, Que. 1, 2, 1, Ans. 1, Que. 26 27, 2, Ans. 1, , 3, 1, , 4, 1, , 5, 4, , 6, 3, , 7, 1, , 8, 2, , 9, 4, , 10 11, 3, 1, , 12 13 14 15 16 17, 1, 1, 1, 3, 1, 3, , 18 19 20, 2, 1, 3, , 21 22 23 24 25, 1, 4, 1, 3, 1, , EXERCISE - 3, , 116