The point  (3,  1) is a point on a circle C with centre  (2,  3) and  C is  orthogonal  to x²+y²=8. The conjugate  point  of (3,  1) w.r. to  x² + y² =8 which lies  on C is

The polar of given point with respect to any one of the circles x²+y²-2kx+c²=0, (k is a variable) always passes through a fixed point whatever be the value of k is

The locus of the foot of the perpendicular drawn from the origin to any chord of the circle x²+y²+2gx+2fy+c=0 which substends a right angle at the origin is

The equation of the circle belonging to the coaxal system of which (2, -3)(0, -4) are the limiting points and passing through the point (2, -1) is

The equation of the circle passing through the points (1, 1), (2, -1), (3, 2) is

If a, b, c are the lengths of tangents from (0, 0) to the circle x²+y²-3x-4y+1=0, x²+y²+4x-6y+4=0, x²+y²-6x-12y+9=0 then the ascending order of a, b, c is

The two circles (x-a)²+(y-b)²=c and (y-b)²+x²=4c have only one real common tangent then

The locus of the point whose shortest distance from the circle x²-2x+6y-6=0 is equal to its distance from the line x-3=0 is

A=(cosθ, sinθ) and B==(sinθ, -cosθ) are two points. The locus of the centroid of ΔOAB where O is the origin is

The conjugate system of the coaxal system x²+y²+2ax+2by+2λ(ax-by)=0, λ is a parameter, is