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.

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

Transverse common tangents are drawn from O to the two circles C_1,C_2 with 4, 2 respectively. Then the ratio of the areas of triangles formed by the tangents drawn from O to the circles C_1 and C_2 and chord of contacts of O w.r.t the circles C_1 and C_2 respectively is

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.

Find the equation of a circle which passes through the point (2,0) and whose centre is the limit of the point of intersection of eth lines 3x+5y=1a n d(2+c)x+5c^2y=1a scvec1.

Two circles C_1a n dC_2 both pass through the points A(1,2)a n dE(2,1) and touch the line 4x-2y=9 at Ba n dD , respectively. The possible coordinates of a point C , such that the quadrilateral A B C D is a parallelogram, are (a ,b)dot Then the value of |a b| is_________

If x_1,x_2,x_3 as well as y_1, y_2, y_3 are in G.P. with same common ratio, then prove that the points (x_1, y_1),(x_2,y_2),a n d(x_3, y_3) are collinear.

If x_1,x_2,x_3 as well as y_1, y_2, y_3 are in G.P. with same common ratio, then prove that the points (x_1, y_1),(x_2,y_2),a n d(x_3, y_3) are collinear.

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

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 circlé of C_2 . Tangents to C_1 , from any point on C_3 intersect C_2 , at P^2 and Q . Find the angle between the tangents to C_2^2 at P and Q. Also identify the locus of the point of intersec- tion of tangents at P and Q .