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 x^2+y^2=a^2 and (x-2a)^2+y^2=a^2 are two equal circles touching each other.

Find the equation of a circle with center (4, 3) touching the circle x^2+y^2=1

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.

Find the equation of the family of circles touching the lines x^2-y^2+2y-1=0.

If the circles (x-a)^(2)+(y-b)^(2)=c^(2) and (x-b)^(2)+(y-a)^(2)=c^(2) touch each other, then

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 .

Find the coordinates of the point at which the circles x^2+y^2-4x-2y+4=0 and x^2+y^2-12 x-8y+36=0 touch each other. Also, find equations of common tangents touching the circles the distinct points.

The point P(-1,0) lies on the circle x^(2)+y^(2)-4x+8y+k=0 . Find the radius of the circle Also determine the equation of the circle of equal radius which touches the given circle at P.

If two circles and a(x^2 +y^2)+bx + cy =0 and p(x^2+y^2)+qx+ry= 0 touch each other, then