A rectangle $PQRS$ is drawn with it four points on the circumference of the circle

Find the largest possible value for the area of the rectangle.

This might give you some ideas: try moving $(x,y)$ around. You might find the largest area from the picture, but can you confirm it using algebra?

What would be the coordinates of the other three points? Can you use them to find an expression for the area of the rectangle in terms of $x$ and $y$? Can you manipulate this using the fact that $x^2+y^2=16$ and find the largest area?

Let $S$ be the area. We get

You can now either use the fact that $(x-y{)}^2\ge 0,$ or substitute to eliminate $y$ from $S=4xy$ and use completing the square. If you choose the second option, it is easier to consider $S^2=16x^2{y}^2$ and then let $u=x^2.$