Slicing Level Surfaces

Question

How do I properly do questions like these? I simply each equation by inputting the value into the variable but after that I am confused as to what I am supposed to do after?

Answer 1

After you replace the variable, you look at the curve in the plane that you get. For instance, in the first case you put $z=1.5$, and you get $$ 2x^2+5y^2+2.25=4, $$ or $$ 2x^2+5y^2=1.75. $$ We may rewrite this as $$ \frac{x^2}{\left(\frac{\sqrt{1.75}}{\sqrt2}\right)^2}+\frac{y^2}{\left(\frac{\sqrt{1.75}}{\sqrt5}\right)^2}=1, $$ where we see that get an ellipse centered at the origin crossing the $x$-axis at $\pm\frac{\sqrt{1.75}}{\sqrt2}\simeq\pm0.94$ and the $y$-axis at $\pm\frac{\sqrt{1.75}}{\sqrt5}\simeq\pm0.6$, which seems to be D in your options.