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 `.

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 .

Tangents are drawn to x^(2)+y^(2)=1 from any arbitrary point P on the circle C_(1):x^(2)+y^(2)-4=0. These tangent meets the circle 'C_(1) 'again in A and B. Locus of point of intersection of tangents drawn to C_(1) at A and B

In Figure, two circles touch each other at the point C . Prove that the common tangent to the circles at C , bisects the common tangent at P and Q .

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

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

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 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 C_1,C_2,a n dC_3 belong to a family of circles through the points (x_1,y_1)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.

Consider the curves C_(1):|z-2|=2+Re(z) and C_(2):|z|=3 (where z=x+iy,x,y in R and i=sqrt(-1) .They intersect at P and Q in the first and fourth quadrants respectively.Tangents to C_(1) at P and Q intersect the x- axis at R and tangents to C_(2) at P and Q intersect the x -axis at S .If the area of Delta QRS is lambda sqrt(2) ,then find the value of (lambda)/(2)