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 diagram of the ellipse and straight line as described in the text

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?

  • What happens to the straight line \(y=x+c\) when we stretch it by a factor of \(\frac12\) in the \(x\)-direction?

  • What happens to the points \(A\), \(B\) and \(M\) when we do this stretch? Is \(M\) still the midpoint of \(AB\) after the stretch?

  • Can you use this information to work out the locus of \(M\) in the stretched version as \(c\) changes (without using any equation for the locus that you have already worked out)?

  • Can you use your answers to work out the equation of the locus of \(M\) in the original problem?

  • What would your answers be if the original straight lines had equations \(y=2x+c\) instead? Or, more generally, \(y=mx+c\) for a fixed value of \(m\)?