Hello,

Originally Posted by

**rajeesh733** 1. (y^2)-2y+5z^2=(x^2)-1

Completing the square for $\displaystyle y$ works well :

$\displaystyle \begin{aligned}y^2-2y+5z^2&=x^2-1 \\

y^2-2y+1-1+5z^2&=x^2-1\\

(y-1)^2+5z^2&=x^2\\

Y^2+Z^2&=X^2

\end{aligned}$

for $\displaystyle X=x$, $\displaystyle Y=y-1$ and $\displaystyle Z=\sqrt{5}z$. What curve is described by $\displaystyle Y^2+Z^2=X^2$ ?

$\displaystyle y=x^2-2x+3\implies (x-1)^2=y-2\implies Y=X^2$ for $\displaystyle X=x-1$, $\displaystyle Y=y-2$ and $\displaystyle Z=z$. The equation $\displaystyle Y=X^2$ doesn't depend on $\displaystyle Z$ so the curve is a cylinder. To find which kind of cylinder it is, note that in the X,Y plane $\displaystyle Y=X^2$ is the equation of a parabola so the curve we're looking for is a parabolic cylinder.