Polar of the origin w.r.t the circle x²+y²+2ax+2by+c=0 touches the circle x²+y²=r² if

The circle orthogonal to the  three circles x²+y²+a_ix+b_iy+c=0, i=1, 2, 3 is

The coaxal system having limiting points (2,3), (-3, 2) is

If the two circles (x-1)²+(y-3)²=r² and x²+y²-8x+2y+8=0 intersect in two distinct points, then

The pole of a straight line with respect to the circle x²+y²=a² lies on the circle x²+y²=9a². If the straight line touches the circle x²+y²=r², then

P(-2, -1) and (0, -3) are the limiting points of a coaxal system of which C= x²+y²+5x+y+4=0 is a member. The circle S= x²+y²-4x-2y-15=0 is orthogonal to the circle C. The point where the polar P cuts the circle S is

A tangent to the circle x²+y²=4 meets the coordinate axes at P and Q. The locus of mindpoint of PQ is

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