If the circles x`^2` + y`^2` + 2ax + c = 0 and x`^2` + y`^2` +2by + c = 0 touch each other, prove that 1/a`^2` + 1 b`^2` = 1/c

Examine whether the circles x ^2 + y ^2 - 2x - 4y = 0 and x ^2 + y ^2 - 8y - 4 = 0 touch each other.

A circle passes through the point of intersection of the circles x ^2 + y ^2 + 13x - 3y = 0 and 2x ^2 + 2y ^2 + 4x - 7y - 25 = 0 and also through the point (1, 1). Find its equation.

If a + b + c = 0 , then prove that 1/(2a^2+bc)+1/(2b^2+ca)+1/(2c^2+ab)=0

If a+b+c=0 , then prove that a^4+b^4+c^4=1/2(a^2+b^2+c^2)^2

Find the value of k for which the line x + 2y + k = 0 touches the circle x ^2 + y ^2 = 5.

If a + b + c = 0 , then prove that (a+b)^2/(ab) + (b+c)^2/(bc) + (c+a)^2/(ca)=3

If the roots of the equation ax^2 + bx + c = 0 are in the ratio 2:3 , then prove that 6b^2 = 25ac

If the equation a(b - c)x^2 + b(c - a)x+c (a -b) = 0 has equal roots, prove that 1/a+1/c=2/b

For what value of p,y = x + p touches the ellipse 2x ^2 + 3y ^2 = 1 ?

If 3x + 2y = -1 , then prove that 27x^3+8y^3+1-18xy = 0