Derive the length of the longest line segment that can be enclosed inside the region A.

Question

Q. Let A be the region in the xy-plane given by

$$A=\{(x,y): x=u+v, y=v, u^2+v^2≤1 \} $$

Derive the length of the longest line segment that can be enclosed inside region A.

My attempt: I found the equation of the region A:

$$x^2+2y^2-2xy-1≤0$$

This is the region enclosed by an ellipse. So the major axis must be the longest line segment. But, I am not able to find the major axis. Please help me out.

Answer 1

Consider the map $f: (u, v) \mapsto (u+v, v)$. This linear map converts from $uv$-coordinates to $xy$-coordinates. If $D$ is the unit circle in the $uv$-plane, then $A = f(D)$ is the ellipse we are looking for. Now the length of the major axis of the eclipse is exactly twice the largest possible length of the image of a unit vector under $f$, so the length you are looking for is exactly $2\|f\|_2$.

Do you know how to find $\|f\|_2$ or should I continue?