Question

Find the sections of the conicoid
x^2/36 + y^2/9 - z^2/4 = x^2-1 by the coordinate and identify them.

Answer 1

Conicoid equation is:$\dfrac{x^2}{6^2}+\dfrac{y^2}{3^2}-\dfrac{z^2}{2^2}=x^2-1$

Axis X: $\dfrac{x^2}{6^2}+\dfrac{0^2}{3^2}-\dfrac{0^2}{2^2}=x^2-1$

$x^2(1- \dfrac{1}{6^2})-1=0$

$x_1=- \dfrac{6}{\sqrt35}$

$x_2=\dfrac{6}{\sqrt35}$

Axis Y:$\dfrac{0}{6^2}+\dfrac{y^2}{3^2}-\dfrac{0^2}{2^2}=0^2-1$

$\dfrac{y^2}{3^2}=-1$

So this conicoid hasn't any common points with axis Y

Axis Z:$-\dfrac{z^2}{2^2}=0^2-1$

$\dfrac{z^2}{2^2}=1$

$z^2=2^2$

$z_1=2$

$z_2=-2$