## Calculus (3rd Edition)

$$x^2+y^2=\left(\frac{ka}{c}\right)^{2} +a^2,$$ $$x^2+y^2=\left(\frac{ka}{c}\right)^{2} -a^2.$$
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 $$x^2+y^2=\left(\frac{ka}{c}\right)^{2} -a^2.$$