More from SUDIPTA GHOSH

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34:, The Circle and the family of circles, If a point in a plane moves such that its distance from a fixed point is always same then, the locus of this moving point is a curve whichis known as circle., The fixed point is known as centre of the circle and this equal distance is known, as radius of the circle., The circle, P,, P5, Here P,, P2,P3.are different positions of point Pand PC = P,C = P2C = P;C .and so on., ...., Fundamental Key Points About a Circle, A, B, A, 20, * Equal chords of a circle are equidistant from the centre and vice-versa., If AB =, CD then OP = OQ., * Angle subtended by any arc at the centre is double to the angle subtended by, any point on remaining arc of the circle., If ZBAC =0 then ZBOC = 20, * The bigger of the two chords of a circle is always nearer to the centre., A, B, B, * Angles in the same segment of a circle are equal., * If two chords int ersect either inside or outside then APx PB = CP x PD., %3D, ACSS.G

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344, The equation of the circle when its center and radius are given :, Let C(h, k) be the center and 'r' is radius then equation of circle is (x - h)2 + (y - k)? = r?-, • If center is at origin i.e. C(h, k) = C(0,0) then equation of circle is x? + y? = r²., %3D, %3D, P(x,y), P(x,y), C(h,k), C(-g,-f), (x-h)² + (y- k)² = r²|, x + y2 + 2gx + 2fy + c 0, The second degree general equation of the circle, The sec ond deg ree general equation ax2 + 2hxy + by +2gx + 2fy + c = 0 represents, a circle if a = b, i.e. cofficient of x? = cofficient of y2 and h, = 0., . x* + y2 + 2gx + 2fy + c 0 represents a circle with center C(-g,-f) and radius 'r'., here r = Vg? +f? –, - ., if g? + f2 - c > 0 i.e. r > 0 then the circle will be real., if g? + f? - c < 0 i.e. r < 0 then the circle will be imaginary., if g? + f2 - c < 0 i.e. r = 0 then the circle will be a point circle., The equation of the circle when its end points of diameters are given :, Let A(x1,y1) and B(x2, y2) are its end po ints of diameter then equation of circle, is (x – x;)(x – X2) + (y - yı)(y - y2) = 0., Parametric Equation, The parametric equation of a curve represents any arbitary point lie on the curve., • The parametric equation of the circle (x - h)+(y - k)= r? is x=h + rcose and y=k+ rsine., AY, P(h+rcos0,k+rsin0), r, P(rcose,rsine), C(h,k), q(0,0), •The parametric equation of the circle x? +y? =r is x = rcose and y= rsine.

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345, Intercepts cut by the circle with coordinate axes., Intercepts made by a circle x +y +2gx +2fy +c 0 on the coordinate axes, is distance, between the two points where the circle cuts the axis of x and axis of y., P(x,y), C(-g,-f), -2g-c>, Intercepts cuts by circle with x-axis =2g? -c and with y-axis =2\f? - c, Tangent and normal at a given point on the circle:, XX,+yy,=r, P(x,,yı), P(x,,y.), 90°, C(0,0), C(-g,-f1), Normal, Any line which touches the circle is known as the tangent of the circle., The normal of a circle is a line which is perpendicular to the tangent at the point of, contact to the circle and always passes through the centre of the circle., Equation of tangent to the circle x + y = r at a point P(x,,y1) is xx, +yy,, „2, = r, y, • Equation of normal to the circle x? + y2 = r² at a point P(x,»y1) is, Y1, Equation of tan gent at a point P(x,,y1) on the circle x? +y? + 2gx + 2fy +c =0 is, XX, + yy, + g(x+x,)+f(y+y1)+c=0, %3D, • Equation of normal at a po int P(x,,y1) on the circle x +y² + 2gx + 2fy+c=0, X- X1, y-y1, is, X1+g, yı +f, Xx,+yy,+g{x+x,)+f[y+y,)+c3D0

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346 JE, Number of tangents drawn to a given circle from a given point : -, The number of tangents drawn from a given po int to a given circle depends on the, position of the point w.r. to the circle., (i) If the point is outside the circle then two tangents of equal length can be drawn., (ii) If the point is on the circle then only one tangent of inf inite length can be drawn., (iii) If the point is inside the circle then no real tangent can be drawn., Pair of tangents : -, Let PQ and PR be two tangents drawn from P(x1,y1) to the circle x? + y? +2gx+2fy +c 0, Then PQ = PR is called the length of tangent drawn from point P and is given by, PQ = PR = xỉ +yỉ + 2gx1 +2fyı+c = /S1., Here PQ, S, and QC =r then,, area of quadrilateral PQCR, 2 Area of APQC, P(x,,y), S,=0, r/S, =r/S,, S., R, Area of Quadrilateral PQCR:, = r /s,, Equations of Pair tangents : -, The combined equations of tangents PQ and PR is SS, =T, here S= x +y +2gx+2fy + c., S = xỉ + yỉ +2gx, +2fy¡ +c, and T= xx1 + yy, +g(x +x1)+f(y+y1)+c, Chord of contacts of tangents :-, The chord joining the points of contact of the two tangents to a circle drawn from, a given point, outside it, is called the chord of contact of tangents., A, P(x,y,), Chord of contact of tangents, B, * The equation of chord of contacts of tangents is same as that of equation of, tangent at that point on the circle., SM

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347, Common tangents of two circles: -, * When two circles touch each other internally :-, Then |C,C2-In- 12| i.e. the distance between the centres is equal to the difference, of the radii., -Tangent, P, P divides C,C, externally in the ratio r : r2, * When two circles touch each other externally :-, The distance between the centres is equal to the sum of radii. C,C2 = rị + r2., %3|, In this case two direct common tangents are real and distinct while the transverse, tangents are coincident., Direct common tan gents, P divides C,C2 externally in the ratio, P, C2, Transverse common tan gent, * When two circles neither touch nor intersect each other :-, Then |C,C,| > r, + r, i.e., the distance between the centres is greater than the, sum of radii. In this case four common tangents can be drawn to the two circles,, in which two are direct common tangents and the other two are transverse common, tangents., Pdivides C,C2 externally in the ratio r, : r,., Direct common tan gents, T divides C,C, internally in the ratio r:r., Length of D.C.T = J(c,C,}? - (r, – r,, %3D, Length of T.C.T = (C,C - (r + r2)², Transverse common tan gents