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| a, , 24.33, qHecIRCLE, 2 yy? -4x-6y=0 I. x? + y? - 6x +12y -15 =0, 40. x, 2 oy? —4x-2y+4=0, x? ty? -2x-4y+4=0, ab 2 2 _, 2 i = 14. | —,-— 15. -4x-10y+25=0, 2 tye -4x 6y+9=0 ($3) x+y ly, , HINTS TO SELECTED PROBLEMS, , circle passes through (0, 0), (4, 0) and (0, 6)., A tae of the given circle is (3, — 6), and radius = ,/9 + 36 —15 = /30. Let r be the radius of, ; the required circle. Then, xr? = 2(n(¥30)*) = r = V60., , 43. The centre of the required circle is (2, 3). As it touches y-axis. So, its radius = x-coordinate of, centre = 2., , 94.5 DIAMETER FORM OF A CIRCLE, , THEOREM The equation of the circle drawn on the straight line joining two given points (x1, Y1) and, (x, Yo) a8 diameter is (x — x4) (x — XQ) +(y—y) (y— Yo) =0., , prOoOr Let A and B be the extremities of the diameter AB having coordinates (x1, yy) and, (x2, Ya) respectively. Let P (x, y) be any point on the circle. Join point P to points A and B.Then,, , a : Y¥-¥2, m, = Slope of the line AP = Y7M". and, my = Slope of the line BP = aoe, x-%y 2, , , , Y, , B(Xy, Yo), , , , , , Fig, 24.38, The angle subtended at the point P in the semi-circle APB is a right angle., mymy = -1, , => ony 4, X-Xy X-Xq, , . (yyy) (Y ~ Yo) =—(% = 4) (X= 2), , S (x= xy) (8 -x9) + (y-yy) (Y- Yo) =0 (i), This is the required equation of the circle having (x1, y,) and (x2, yy) as the coordinates of the, end points of a diameter. QED., REMARK 1, , , , If the coordinates of the end points of a diameter of a circle are given, we can also find the, , equation of the circle by finding the coordinates of the centre and radius. The centre is the mid-point of the, diameter and radius is half of the length of the diameter., , Dien, , Scanned with CamScanner

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eo 24.37, , we LAOB = ZBOC = ZCOD = ZDOA & n/2. Therefore, AB, BC, CD and DA are diameters, ane four circles., ZXOA = a om OA@a, , Now nm a, ‘ vg I, AC, =4 sin = 1 + and, OC) =a cos "vi, gp, the coordinates of A are (a//2, a/V2),, , siti the coordinates of B, C and D are(-a/ V2, a//2), (- a//2,-a/ /2) and (a//2, ~ a//2), respectively ly., , The equation of the circle with AD as diameter is, , a a a a ~, (-#) (: “3 [y- ally fi) = 0 of, x2 4+ y? V2 ax = 0,, , simile arly, the equations of the re me circles with BC, CD and - as diameters are, , a a, bE la) Oran a) tm test odtaco, , —m_~, <, 4, , #) y+) = 0 or, x2 4+y* + V2 ay=0, , oA: [gente ee fag BL = a a, , respectively., , EXERCISE 24.3, , , , , , LEVEL-1, , , , , , , , 1. Find the equation of the circle, the end points of whose diameter are (2, — 3) and (— 2, 4)., Find its centre and radius., , 2 Find the equation of the circle the end points of whose diameter are the centres of the, circles x? + y + 6x- 14y -1 = Oand x* + yy - 4x+ 10y — 2=0., , 3. The sides of a square are x =6, x=9, y= 3and y =6. Find the equation of a circle drawn, on the diagonal of the square as its diameter., , 4. Find the equation of the circle circumscribing the rectangle whose sides are x — 3y = 4,, 3x+y=22,x-3y=l4and 3x + y = 62., , 5. Find the equation of the circle passing through the origin and the points where the line, , 3x + 4y =12 meets the axes of coordinates., , Find the equation of the circle which passes through the origin and cuts off intercepts a and, , b respectively from x and y-axes., , Find the equation of the circle whose diameter is the line segment joining (— 4, 3) and, , (12, 1). Find also the intercept made by it on y-axis., , LEVEL-2_, , , , , , , , , , 8. The abscissae of the two points A and B are the roots of the equation x? + 2ax-b? =Oand, their ordinates are the roots of the equation x? + 2px - a = 0. Find the equation of the circle, with AB as diameter. Also, find its radius., , : ABCDisa square whose side is 4; taking BAB and AD as axes, prove that the equation of the, circle circumscribing the square is xy e ~a(x+y)=0., , 10, The line 2x — y + 6=0 meets the circle x 4 y ~2y -9 =Oat A and B. Find the equation of, , the circle on AB as diameter., , Scanned with CamScanner

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24.38 Yv, , ads, , MATHEM, ATIeg, |, , Find the equation of the circle which circumscribes the trian;, , |, x=0, y=Oand Ix + my =1, Ble formed ly, , the lines, , , , 12. Find the equations of the circles which igi, ind th pass through the origin and cut, 2 units from the lines y = x and y =-x. nega hotdog, ANSWey, Rg, 1. P+y¥-y-16-0;(0,3), 2x2 4y? +x-2y-41=0, 3. x? 497 -15r-9y4+72=0 4.x? 4? -27x-3y+142=0, 5. x2 4 y? -4x-3y=0 6 x7 +y? taxtby =0, 7. x4? ~8x-2y-51=0, 4v13, 8. x2 + y? + 2ax + 2py -b? - 42 =0, Va? +b? +p, 10. x2 44? 4+4x-4y+3=0 11. ay-tr-lyso, m, 12 x? 4y? +2y=0, x2 44? +2x=0., HINTS TO SELECTED PROBLEns, , , , ®, , 6., 9,, , The line 3x + 4y = 12 meets the coordinate axes at A (4, 0) and B(0, 3). We have to find the, equation of the circle with AB as diameter., , The coordinates of the end points of a diameter are (+ a, 0) and (0, + b)., The required circle has (0, 0) and (a, a) as the end points of a diameter., MULTIPLE CHOICE QUESTIONS (MCQs), , , , Mark the correct alternatives in each of the following:, 1., , If the equation of a circle is Ax? +(24= 3) y —4x + 6y—1=0, then the coordinates of, centre are, , (a) (4/3, -1) (6) (2/3,-1) (c) (-2/3, 1) (d) (2/3,1), , , , 2 fax? 4 Lay + 2y? +(,—4) x + 6y —5 = Dis the equation of a circle, then its radius is, (a) 3 v2 (b) 23 (c) 2V2 (d) none of these, 3. The equation x4 y +2x-4y+5=0 represents |, (a) a point (b) a pair of straight lines |, (c) acirele of non-zero radius (d) none of these |, 4. If the equation (4a ~ 3) v4 ay? + 6x —2y + 2 =O represents a circle, then its centre is, (a) (3, -D (b) (3,1) (c) (- 3,1) (d) none of these, 5. The radius of the circle represented by the equation 3x7 + 3y? +hxy + 9x +(h- 89, +3=0 is, (a) 3/2 (b) VI7/2 (c) 2/3 (4) noneof these, 6. The number of integral values of 2. for which the equation Payr+ax+(1-Nyt5*, » equati f a circle whose radius cannot exceed 5, is, he , “ ala 18 (c) 16 (d) none of these and, , : i jameters, 7. The equation of the circle passing through the point (1, 1) and having two diame, , the pair of lines x7 - y* - 2x + dy - 3 =0.is |, ({b) vty? +2x+4y-4=0, , (d) none of these, , (a) x2 +y? -2x-4y +4 =O, , (c) xP gy? -2x+dy+4=0, , Scanned with CamScanner

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re, , , , 39, qHE CIRCLE 243, , 8, , 10., , 11., , 13., , 14., , a5:, , 16., , 17,, 18., 19,, , 20., , If the centroid of an equilateral triangle is (1, 1) and its one vertex is (—1, 2), then the, equation ot its circumcircle is, , Ok vv -2x-2y-3=0 (b) x7 +y? +2x-2y-3=0, (2+ y? +2x+2y-3=0 (A) none of these, . If the point (2, k) lies outside the circles xr 4 v7 +x-2y-14 50 and x4 y7 =13 then k lies, in the interval |, (a) (- 3,=2) U(3, 4) (b) =, () © 21-3) U (4, ®) (d) (— 9, - 2) U(3, »), , If the point (1, +1) lies inside the region bounded byt the curve x54 25 > ay ang y-axis,, then belongs to the interval |, , (a) (-1, 3) (b) (4, 3), (c) (- 2%, - 4) U(3, ~) (d) none of these, The equation of the incircle formed by the coordinate axes and the line 4x + 3y = 6 is, (a) 7 +y? -6x-6y+9 = 0 . AG? eye -x-y) +1 =, (4(t+ytxt+y 41 =0 " @) none of these, , Ifthe circles x? +y?=9 and x? + y2+8 y+¢=O0touch each other, then cis equal to, (a). 15 (b) -15 1) (c) 16 (a) -16, If the circle x” + y? + 2ax +8 y +16 =0 touches x-axis, then the value of a is, (a) +16 (b) +14 (c) +8 (d) +1, The equation of a circle with radius 5 and touching both the coordinate axes is, (a) x+y? +10x+10y+5=0 (b) x7 +y? £10 x+10y=0, () 2 +y? £10x4+10y+25=0 (a) x2. +4? +10 x+10 y+51=0, The equation of the circle passing through the origin which cuts off intercept of length 6, and 8 from the axes is, (a) x7 +¥° -12x-16y=0 (b) x7 +y? 412x416 y=0, ( +y?4+6x4+8y=0 i Se anne 8y=0, , The equation of the circle concentric with x? + y? —3x+4y—-c=Oand pa: ing through) |, CL- Bee ve, , (a) x? +y ~3x44y-1=0))))! (b) x7 4y2-3x44y=0, , (c) oy —-3x44y4+2=0 (d) none of these, , Peete x Ait aero ceMnna tnt not intersect x-axis, if, , (a) g<c (b) gs > e)i)1)) (c) gr >2e | a none of these, , The area of an equilateral tangle inscribed in the circle x? + y -6x-8y-25=0is, , @) a, , , , (b) 35 no i (c) 502 100) (d) none of these, The equation of the circle which touches the axes of coordinates and the line . +4=1and, , whose centres lie in the first quadrant is x? + y?—2cx-2 cy +c? =0, where cis equal to, (a) 4 3”) 2 | () 3 (d) 6, , If the circles x2 + y =a and x? xe + y* -6x-8y+9=0, touch externally, then a =, , (a) 1° (b) -1 (©) 21 (d) 16, , Scanned with CamScanner