Prove that the equation of the chord of the ellipse \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\] joining the points \((a\cos\alpha,b\sin\alpha)\) and \((a\cos\beta,b\sin\beta)\) is \[\frac{x}{a}\cos\left(\frac{\alpha+\beta}{2}\right)+\frac{y}{b}\sin\left(\frac{\alpha+\beta}{2}\right)= \cos\left(\frac{\alpha-\beta}{2}\right).\]

What’s the equation of a line passing through two points whose coordinates we know?

Could we look up and use the standard formulae for \(\sin x-\sin y\), and \(\cos x-\cos y\)?

Through a point \(P\) on the major axis of an ellipse a chord \(HK\) is drawn. Prove that the tangents at \(H\) and \(K\) meet the line through \(P\) at right angles to the major axis at points equidistant from \(P\).

Try varying \(\alpha, \beta, a\) and \(b\) here. The question asks us to show that \(PH' = PK'\) (that the light green length equals the yellow length).

Does your experimenting back this up? Can we now turn this into a proof?

What are the coordinates of \(P\)?

How do we find the gradient of a tangent to an ellipse? What’s the equation of the tangent through \(H\)?

What are we trying to prove about the \(y\)-coordinates of \(H'\) and \(K'\)?