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

The line 4x+2y = c is a tangent to the parabola y^(2) = 16x then c is :

The circle x^2 + y^2 - 4x + 6y + c = 0 touches x axis if

If the line y=mx+c is a tangent to the ellipse x^(2)+2y^(2)=4 , find c

Let S be the circle in the x y -plane defined by the equation x^2+y^2=4. (For Ques. No 15 and 16) Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N . Then, the mid-point of the line segment M N must lie on the curve (a) (x+y)^2=3x y (b) x^(2//3)+y^(2//3)=2^(4//3) (c) x^2+y^2=2x y (d) x^2+y^2=x^2y^2

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 .

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

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 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+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