Prove that the circle `x^2 + y^2 + 2ax + c^2 =0 and x^2 + y^2 + 2by + c^2 = 0` touch each other if `1/a^2 + 1/b^2 = 1/c^2`.

Two circles centres A and B radii r_1 and r_2 respectively. (i) touch each other internally iff |r_1 - r_2| = AB . (ii) Intersect each other at two points iff |r_1 - r_2| ltAB lt r_1 r_2 . (iii) touch each other externally iff r_1 + r_2 = AB . (iv) are separated if AB gt r_1 + r_2 . Number of common tangents to the two circles in case (i), (ii), (iii) and (iv) are 1, 2, 3 and 4 respectively. circles x^2 + y^2 + 2ax + c^2 = 0 and x^2 + y^2 + 2by + c^2 = 0 touche each other if (A) 1/a^2 + 1/b^2 = 2/c^2 (B) 1/a^2 + 1/b^2 = 2/c^2 (C) 1/a^2 - 1/b^2 = 2/c^2 (D) 1/a^2 - 1/b^2 = 4/c^2

Prove that the circle x^(2)+y^(2)+2x+2y+1=0 and circle x^(2)+y^(2)-4x-6y-3=0 touch each other.

Prove that the circles x^(2)+y^(2)+2ax+ay-3a^(2)=0andx^(2)+y^(2)-8ax-6ay+7a^(2)=0 touch each other.

The condition that the circles x^(2)+y^(2)+2ax+c=0, x^(2)+y^(2)+2by+c=0 may touch each other is

The condition that the circles x^(2)+y^(2)+2ax+c=0, x^(2)+y^(2)+2by+c=0 may touch each other is

Prove that the circles x^2+y^2+24ux+2vy=0 and x^2+y^2+2u_1x+2v_1y=0 touch each other externally if -12u_1u=v_1v .

Prove that the circle x^2 + y^2 =a^2 and (x-2a)^2 + y^2 = a^2 are equal and touch each other. Also find the equation of a circle (or circles) of equal radius touching both the circles.

Prove that the circle x^2 + y^2 =a^2 and (x-2a)^2 + y^2 = a^2 are equal and touch each other. Also find the equation of a circle (or circles) of equal radius touching both the circles.

If the circles x^2+y^2+2ax+c=0 and x^2+y^2+2by+c=0 touch each other, then find the relation between a, b and c .

The circles x^(2)+y^(2)+2x-2y+1=0 and x^(2)+y^(2)-2x-2y+1=0 touch each other