If `C_1,C_2,a n dC_3` belong to a family of circles through the points `(x_1,y_2)a n d(x_2, y_2)` prove that the ratio of the length of the tangents from any point on `C_1` to the circles `C_2a n dC_3` is constant.

If the length of the tangent from (f,g) to the circle x^(2)+y^(2)=6 be twice the length of the tangent from the same point to the circle x^(2)+y^(2)+3x+3y=0, then

Find the length of the tangent drawn from any point on the circle x^2+y^2+2gx+2fy+c_1=0 to the circle x^2+y^2+2gx+2fy+c_2=0

If the distances from the origin of the centers of three circles x^2+y^2+2lambdax-c^2=0,(i=1,2,3), are in GP, then prove that the lengths of the tangents drawn to them from any point on the circle x^2+y^2=c^2 are in GP.

Consider three circles C_1, C_2 and C_3 such that C_2 is the director circle of C_1, and C_3 is the director circle of C_2 . Tangents to C_1 , from any point on C_3 intersect C_2 , at P and Q . Find the angle between the tangents to C_2 at P and Q. Also identify the locus of the point of intersec- tion of tangents at P and Q .

The chords of contact of tangents from three points A ,Ba n dC to the circle x^2+y^2=a^2 are concurrent. Then A ,B a n dC will be

C_1 and C_2 are circle of unit radius with centers at (0, 0) and (1, 0), respectively, C_3 is a circle of unit radius. It passes through the centers of the circles C_1a n dC_2 and has its center above the x-axis. Find the equation of the common tangent to C_1a n dC_3 which does not pass through C_2dot

If the line joining the points (0,3)a n d(5,-2) is a tangent to the curve y=C/(x+1) , then the value of c is 1 (b) -2 (c) 4 (d) none of these

Two circles C_1a n dC_2 intersect at two distinct points Pa n dQ in a line passing through P meets circles C_1a n dC_2 at Aa n dB , respectively. Let Y be the midpoint of A B ,a n dQ Y meets circles C_1a n dC_2 at Xa n dZ , respectively. Then prove that Y is the midpoint of X Zdot

If C_(1),C_(2),C_(3),... is a sequence of circles such that C_(n+1) is the director circle of C_(n) . If the radius of C_(1) is 'a', then the area bounded by the circles C_(n) and C_(n+1) , is

Find the angle between the curves x^2-(y^2)/3=a^2a n dC_2: x y^3=c