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 at`X_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 a line L_(1):3x+2y-6=0 intersect the x and y axes at P and Q respectively.Let another line L_(2) perpendicular to L_(1) cut the x and y axes at R and S respectively.The locus of point of intersection of the lines PS and QR is

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.

l_(1) and l_(2) are the side lengths of two variable squares S_(1), and S_(2), respectively.If l_(1)+l_(2)^(3)+6 then rate of change of the area of S_(2, with respect to rate of change of the area of ) of S_(1) when l_(2)=1 is equal to

The equation of a circle S_1 is x^2 + y^2 = 1 . The orthogonal tangents to S_1 meet at another circle S_2 and the orthogonal tangents to S_2 meet at the third circle S_3 . Then (A) radius of S_2 and S_1 are in ratio 1 : sqrt(2) (B) radius of S_2 and S_1 are in ratio 1 : 2 (C) the circles S_1, S_2 and S_3 are concentric (D) none of these

Let a given line L_(1) intersect the X and Y axes at P and Q respectively.Let another line L_(2) perpendicular to L_(1) cut the X and Y -axes at Rand S,respectively.Show that the locus of the point of intersection of the line PS and QR is a circle passing through the origin

Let L_1 be a tangent to the parabola y^(2) = 4(x+1) and L_2 be a tangent to the parabola y^(2) = 8(x+2) such that L_1 and L_2 intersect at right angles. Then L_1 and L_2 meet on the straight line :

C_1 is a curve y^2=4x,C_2 is curve obtained by rotating C_1,120^@ in anti -clockwise direction C_3 is reflection of C_2 with respect to y=x and S_1,S_2,S_3 are foci of C_1,C_2 and C_3 , respectively, where O is origin. IF S_1(x_1,y_2),S_2(x_2,y_2) and S_3(x_3,y_3) then the value of sumx_1^2+sumy_1^2 is

A given line L_1 cut x and y-axes at P and Q respectively and has intercepts a and b/2 on x and y-axes respectively. Let another line L_2 perpendicular to L_1 cut x and y-axes at R and S respectively. Let T be the point of intersection of PS and QR . Locus of T is a circle having centre at (A) (a, b) (B) (a, b/2) (C) (a/2, b) (D) (a/2, b/4)