Answer
$$ x^2+y^2=\left(\frac{ka}{c}\right)^{2} +a^2,$$
$$ x^2+y^2=\left(\frac{ka}{c}\right)^{2} -a^2.$$
Work Step by Step
The equation of a hyperboloid of one sheet is
$$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1+\left(\frac{z}{c}\right)^{2} .$$
To find the horizontal traces, we put $ z=k $; hence the equation becomes
$$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1+\left(\frac{k}{c}\right)^{2} .$$
For this equation to be a circle, we must have $ a=b $ and hence
$$ x^2+y^2=a^2+\left(\frac{ka}{c}\right)^{2} .$$
Similarly, for the hyperboloid of two sheets, we get
$$ x^2+y^2=\left(\frac{ka}{c}\right)^{2} -a^2.$$
