The condition that the circles x²+y²+2ax+c=0, x²+y²+2by+c=0 may touch each other is

If the circles described on the line joining the points (0, 1) and (α, β) as diameter cuts the axis of x in points whose abscissa are the roots of the equation x²-5x+3=0, then (α, β)=

The point of intersection of the tangents to the circle passing through (4, 7), (5, 6). (1, 8) at the points where it is cut by the line 5x+y+17=0 is

The centres of the three circles x²+y²-10x+9=0, x²+y²-6x+2y+1=0, x²+y²-9x-4y+2=0 lie on the line

If y=4x-5 is a tangent to the curve y²=px³+q at (2, 3), then

For the circles x²+y²+4x+2y-4=0, x²+y²-4x-2y+4=0 the line 3x+4y-5=0 is a

The locus of midpoints of the chord of the circle x²+y² = 25 which pass through a fixed point (4, 6) is a circle. The radius of that circle is

If one end of the diameter of the circle x²+y²-6x+4y-12=0 is (7, -5),  then the other end of the diameter is

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