Show that the centres of the following circles lie on a line and their radii are in A.P. : `x^(2) + y^(2) = 1, x^(2) + y^(2) + 6x - 2y -6 = 0, x^(2) + y^(2) - 12x + 4y - 9 = 0`.

The circle x^2 + y^2 - 2x - 4y + 1 = 0 and x^2 + y^2 + 4x + 4y - 1 = 0

Prove that the radii of circles x^2 + y^2 = 1, x^2 + y^2 - 2x -6y = 6 and x^2 + y^2 - 4x - 12y = 9 are in A.p.

Prove that the centres of the circles x^(2) + y^(2) - 10x + 9 = 0, x^(2) + y^(2) - 6x + 2y + 1 = 0 and x^(2) + y^(2) - 18x - 4y + 21 = 0 lie on a line, find the equation of the line on which they lie.

Prove that the centres of the three circles x^(2) + y^(2) - 2x + 6y + 1 = 0, x^(2) + y^(2) + 4x - 12y + 9 = 0 and x^(2) + y^(2) - 16 = 0 are collinear.

Find the equation to the common chord of the two circles x^(2) + y^(2) - 4x + 6y - 36 = 0 and x^(2) + y^(2) - 5x + 8y - 43 = 0 .

Let P be the point on the parabola, y^2=8x which is at a minimum distance from the centre C of the circle, x^2+(y+6)^2=1. Then the equation of the circle, passing through C and having its centre at P is : x^ 2 + y^ 2 − x + 4 y + 12 = 0 x^ 2 + y^ 2 − x/ 4 + 2 y − 24 = 0 x^ 3 + y^ 2 − 4 x + 9 y − 18 = 0 x^ 2 + y^ 2 − 4 x + 8 y + 12 = 0

In each the following find the centre and radius of circles. x^(2)+y^(2)-8x+10y-12=0 .

In each the following find the centre and radius of circles. x^(2)+y^(2)-4x-8y-45=0 .

Show that the circles x^(2) + y^(2) + 6x + 2y + 8 = 0 and x^(2) + y^(2) + 2x + 6y + 1 = 0 intersect each other.

The locus of the centre of the circle which bisects the circumferences of the circles x^2 + y^2 = 4 & x^2 + y^2-2x + 6y + 1 =0 is :