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 \[\frac{x}{a}\cos\left(\frac{\alpha+\beta}{2}\right)+\frac{y}{b}\sin\left(\frac{\alpha+\beta}{2}\right)= \cos\left(\frac{\alpha-\beta}{2}\right).\]
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 \(P\).
