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

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

Prove that the curves y^2=4x and x^2+y^2-6x+1=0 touch each other at the points (1,\ 2) .

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

Let f(x) be a differentiable function and f(x)>0 for all x. Prove that the curves y=f(x) and y=sin kxf(x) touch each other at the points of intersection of the two curves.

Show that the curves xy=a^(2) and x^(2)+y^(2)=2a^(2) touch each other

Prove that the curves 2y^(2)=x^(3) and y^(2)=32x cut each other orthogonally at the origin

Prove that the curve y^(2)=4x and x^(2)+y^(2)-6x+1=0 touches each other at thepoint (1,2), find the equation of the common tangents.

Prove that the curve x^(2)+y^(2)=1 and y^(2)=4(x-1) touches each other.

Prove that the curves x^(2)-y^(2)=16 and xy = 15 intersect each other at 90^(@) angle.

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.