### Circles

Package of problems

## A certain point of view

This diagram shows the ellipse with equation $x^2+4y^2=16$ together with the straight line $y=x+c$.

The line and ellipse intersect at the points $A$ and $B$, and the midpoint of $AB$ is $M$.

Find the coordinates of $M$ for some different values of $c$.

What do you notice?

What is the locus of $M$ as $c$ varies?

What would happen if you used straight lines of a different gradient instead?

The ellipse looks like a stretched circle: it is the circle $x^2+y^2=4$ stretched by a factor of $2$ in the $x$-direction.

What happens to the problem when we undo the stretch, by stretching by a factor of $\frac12$ in the $x$-direction?