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?