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 C1 ​ but outside C2 ​ 2)P lies inside C2 ​ but outside C1 ​ c)P lies outside C1 ​ but outside C2 ​ d)P does not lie inside C2 ​ List 2 p)λ ϵ (−∞,−1)∪(0,∞) q)λ ϵ (−∞,−1)∪(1,∞) r)λ ϵ (−1,0) s)λ ϵ (0,1)

The line L_(1)-=3x-4y+1=0 touches the circles C_(1) and C_(2) . Centers of C_(1) and C_(2) are A_(1)(1, 2) and A_(2)(3, 1) respectively Then, the length (in units) of the transverse common tangent of C_(1) and C_(2) is equal to

If the lengths of the tangents drawn from P to the circles x^(2)+y^(2)-2x+4y-20=0 and x^(2) +y^(2)-2x-8y+1=0 are in the ratio 2:1, then the locus P is

Let C_1 and C_2 be parabolas x^2 = y - 1 and y^2 = x-1 respectively. Let P be any point on C_1 and Q be any point C_2 . Let P_1 and Q_1 be the reflection of P and Q, respectively w.r.t the line y = x then prove that P_1 lies on C_2 and Q_1 lies on C_1 and PQ >= [P P_1, Q Q_1] . Hence or otherwise , determine points P_0 and Q_0 on the parabolas C_1 and C_2 respectively such that P_0 Q_0 <= PQ for all pairs of points (P,Q) with P on C_1 and Q on C_2

Let C_1 and C_2 be respectively, the parabolas x^2=y-1 and y^2=x-1 Let P be any point on C_1 and Q be any point on C_2 . Let P_1 and Q_1 be the refelections of P and Q, respectively with respect to the line y=x. If the point p(pi,pi^2+1) and Q(mu^2+1,mu) then P_1 and Q_1 are

Let C_1 and C_2 be respectively, the parabolas x^2=y-1 and y^2=x-1 Let P be any point on C_1 and Q be any point on C_2 . Let P_1 and Q_1 be the refelections of P and Q, respectively with respect to the line y=x. Arithemetic mean of PP_1 and Q Q_1 is always less than

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 the tangent to the circle x^(2)+y^(2)-2x-y-7=0 from (-1, -3), is

Consider the circle x^(2)+y^(2)_4x-2y+c=0 whose centre is A(2,1). If the point P(10,7) is such that the line segment PA meets the circle in Q with PQ=5 then c=

Let the line segment joining the centres of the circles x^(2)-2x+y^(2)=0 and x^(2)+y^(2)+4x+8y+16=0 intersect the circles at P and Q respectively. Then the equation of the circle with PQ as its diameter is