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 `.

if a variable tangent of the circle x^(2)+y^(2)=1 intersects the ellipse x^(2)+2y^(2)=4 at P and Q. then the locus of the points of intersection of the tangents at P and Q is

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

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 the tangent to the ellipse x^(2)+2y^(2)=1 at point P((1)/(sqrt(2)),(1)/(2)) meets the auxiliary circle at point R and Q, then find the points of intersection of tangents to the circle at Q and R.

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)

In the given figure (not to scale), two circles C_1 and C_2 intersect at S and Q. PQN and RQM are tangents drawn to C_1 and C_2 respectively at Q. MAB and ABN are the chords of the circles C_1 and C_2 . If angle NQR =85^@ , then find angle AQB .

C_(1) and C_(2) are two concentrate circles, the radius of C_(2) being twice that of C_(1) . From a point P on C_(2) tangents PA and PB are drawn to C_(1) . Prove that the centroid of the DeltaPAB lies on C_(1)

A line y=2x+c intersects the circle x^(2)+y^(2)-2x-4y+1=0 at P and Q. If the tangents at P and Q to the circle intersect at a right angle,then |c| is equal to