The condition that the circles (x-α)²+(y-β)²=r², (x-β)²+(y-α)²=r² may touch each other is

The length of the chord of contact of (-2, 3) with respect to the circle x²+y²-2x+4y+1=0 is

From the origin chords are drawn to x²+y²-2y=0. The locus of the middle points of these chords is

Consider a family of circles which are passing through the point (-1, 1) and are tangent to x-axis. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of K is given by the interval

The locus of the point of intersection of two tangents drawn to the circle x²+y²=a² which make a constant angle α to each other is

The straight line x-2y+1=0intersects the line circle x²+y²=25 in points P and Q, the coordinates of the point of intersection of tangents drawn at P and Q to the circle is

If the tangents at (3, -4) to the circle x²+y²-4x+2y-5=0 w.r.t the circle x²+y²+16x+2y+10=0 in A and B, then the midpoint of AB is

The polar of (x₁, y₁) w.r.t the circle x²+y²=a² meets the coordinate axes in A and B. The area of ΔOAB is

The equations of the tangents drawn from the origin x²+y²+2gx+2fy+f²=0 is

The area of the triangle formed by the tangents from (1, 3) to the circle x²+y²-4x+6y+1=0 and its chord of contact is