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

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.

Show that the curves x y=a^2 and x^2+y^2=2a^2 touch each other.

Consider the circles x^(2)+(y-1)^(2)=9,(x-1)^(2)+y^(2)=25. They are such that these circles touch each other one of these circles lies entirely inside the other each of these circles lies outside the other they intersect at two points.

Statement 1 : The number of common tangents to the circles x^(2) +y^(2) -x =0 and x^(2) +y^(2) +x =0 is 3. Statement 2 : If two circles touch each other externally then it has two direct common tangents and one indirect common tangent.

If circles (x -1)^(2) + y^(2) = a^(2) " and " (x + 2)^(2) + y^(2) = b^(2) touch each other externally , then

Statement 1 : The number of common tangents to the circles x^(2) + y^(2) =4 and x^(2) + y^(2) -6x - 6y = 24 is 3. Statement 2 : If two circles touch each other externally thenit has two direct common tangents and one indirect common tangent.

Prove that the curves x y=4 and x^2+y^2=8 touch each other.

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 .