Let `C_1 and C_2` are circles defined by `x^2+y^2 -20x+64=0` and `x^2+y^2+30x +144=0`. The length of the shortest line segment PQ that is tangent to `C_1` at P and to `C_2` at Q is

Let C_1 and C_2 be two circles whose equations are x^2+y^2-2x=0 and x^2+y^2+2x=0 and P(lambda, lambda) is a variable point . List 1 a) P lies inside C_1 ​ but outside C_2 ​ b)P lies inside C_2 ​ but outside C_1 ​ c)P lies outside C_1 ​ but outside C_2 ​ d)P does not lie inside C_2 ​ List 2 p)λ ϵ (−∞,−1)∪(0,∞) q)λ ϵ (−∞,−1)∪(1,∞) r)λ ϵ (−1,0) s)λ ϵ (0,1)

Let C_(1) and C_(2) denote the centres of the circles x^(2) +y^(2) = 4 and (x -2)^(2) + y^(2) = 1 respectively and let P and Q be their points of intersection. Then the areas of triangles C_(1) PQ and C_(2) PQ are in the ratio _

If the straight line x - 2y + 1 = 0 intersects the circle x^2 + y^2 = 25 at points P and Q, then find the coordinates of the point of intersection of the tangents drawn at P and Q to the circle x^2 + y^2 = 25 .

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.

Find the condition if the circle whose equations are x^2+y^2+c^2=2a x and x^2+y^2+c^2-2b y=0 touch one another externally.

If the circles x^2+y^2+2ax+c^2=0 and x^2+y^2+2by+c^2=0 touch each other,prove that 1/a^2+1/b^2=1/c^2

If the circles x^2+y^2+2a x+c y+a=0 and x^2+y^2-3a x+d y-1=0 intersects at points P and Q , then find the values of a for which the line 5x+b y-a=0 passes through Pa n dQdot

Let x^2 +y^2 -2x-2y-2=0 and x^2 +y^2 -6x-6y+14=0 are two circles C_1, C_2 are the centre of circles and circles intersect at P,Q find the area of quadrilateral C_1 P C_2 Q (A) 12 (B) 6 (C) 8 (D) 4

if the distances from the origin to the centre of three circles x^2+y^2+2lambda_ix-c^2=0(i=1,2,3) are in G.P. ,then the lengths of the tangents drawn to them from any point on the circle x^2+y^2=c^2 are in