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 would be a simple value of \(c\) to choose first?

If you have found an \(x\) or \(y\) value, which equation is simplest to use to find the other one?

Remember that the coordinates of \(M\) are the average (mean) of the coordinates of \(A\) and \(B\).

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