Two circles are `S_1 = (x + 3)^2 + y^2 = 9,S_2 = (x-5)^5 + y^2 = 16` with centres `C_1, & C_2`. A direct common tangent is drawn from a point P (on x-axis) which touches `S_1, & S_2` at `Q & R`, respectively.Find the ratio of area of `DeltaPQC_1, & DeltaPRC_2`.

Consider the circle x^(2) + y^(2) = 9 . Let P by any point lying on the positive x-axis. Tangents are drawn from this point to the given circle, meeting the Y-axis at P_(1) and P_(2) respectively. Then find the coordinates of point 'P' so that the area of the DeltaPP_(1)P_(2) is minimum.

Consider two circles S_1=x^2+(y-1)^2=9 and S_2=(x-3)^2+(y-1)^2=9 , now for these circles match the following :

Consider two circles S, =x^2+y^2 +8x=0 and S_2=x^2+y^2-2x=0 . Let DeltaPOR be formed by the common tangents to circles S_1 and S_2 , Then which of the following hold(s) good?

For the circles S_1: x^2 + y^2-4x-6y-12 = 0 and S_2 : x^2 + y^2 + 6x + 4y-12=0 and the line L.:x+y=0 (A) L is common tangent of S_1 and S_2 (B) L is common chord of S_1 and S_2 (C) L is radical axis of S_1 and S_2 (D) L is perpendicular to the line joining the cente of S_1 & S_2

Two circle S_1 and S_2 with centre C_1&C_2 intersect at X&Y. Secants L_1&L_2 are drawn to S_1&S_2 to intersect atX_1,Y_1 and X_2,Y_2 rwspectively. If L_1&L_2 intersect at Z in exterior ragions of S_1 and S_2 and ZX_1 ZY_1 = ZX_2 ZY_2, then

Let the circles S_(1)-=x^(2)+y^(2)+4y-1=0 S_(2)-= x^(2)+y^(2)+6x+y+8=0 touch each other . Also, let P_(1) be the point of contact of S_(1) and S_(2) , C_(1) and C_(2) are the centres of S_(1) and S_(2) respectively. The coordinates of P_(1) are

Let S_(1): x^(2)+y^(2)=9 and S_(2) : (x-2)^(2)+y^(2)=1 . Then the locus of centre of a variable circle S which touches S_1 internally and S_2 externally always passes through the points:

The circle C1 : x^2 + y^2 = 3 , with center at O, intersects the parabola x^2 = 2y at the point P in the first quadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3, respectively. Suppose C2 and C3 have equal radii 2sqrt(3) and centers Q2 and Q3, respectively.If Q_2 and Q_3 lies on the y-axis, then