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.

The length of tangent from the point (2,-3) to the circle 2x^(2)+2y^(2)=1 is

If the distances from the origin of the centers of three circles x^(2)+y^(2)+2 lambda x-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.

Show that the length of the tangent from anypoint on the circle : x^2 + y^2 + 2gx+2fy+c=0 to the circle x^2+y^2+2gx+2fy+c_1 = 0 is sqrt(c_1 -c) .

If circle C passing through the point (2, 2) touches the circle x^2 + y^2 + 4x - 6y = 0 externally at the point (1, 1), then the radius of C is:

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 a circle C passing through the point (4,0) touches the circle x^(2)+y^(2)+4x-6y=12 externally at the point (1,-1) ,then the radius of C is

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