Functions of Several Variables — Question 2

Question 2 – Solution

Strategy Treat $c$ by sign and normalize the positive-level equation.

Step 1: Cases Because squares are nonnegative, $c<0$ gives no points. For $c=0$, both squares vanish, giving $(0,0)$. For $c>0$, $$\frac{x^2}{c}+\frac{y^2}{c/4}=1.$$ Thus the semiaxes are $\sqrt c$ and $\sqrt c/2$.

Step 2: Area The enclosed area is $$A(c)=\pi\left(\sqrt c\right)\left(\frac{\sqrt c}{2}\right)=\boxed{\frac{\pi c}{2}}.$$

Step 3: Graph Higher levels produce nested, expanding ellipses centered at the origin. Since $z=x^2+4y^2\ge 0$ and increases quadratically, the graph is an upward elliptic paraboloid, steeper in the $y$-direction.