The point \(P\) in the first quadrant lies on the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). The points \(A^{\prime} \ (-a, 0)\) and \(A \ (a, 0)\) are the extremities of the major axis of the ellipse, and \(O\) is the origin. The tangent to the ellipse at \(P\) meets the axis of \(y\) at \(Q\) and meets the line \(x = a\) at \(T\). The chord \(A^{\prime} P\) meets the axis of \(y\) at \(M\), and when produced meets the line \(x=a\) at \(R\).

Prove that

  1. \(AT=TR\),

  2. \(OQ^2 - MQ^2 = b^2\).

How can we find the tangent to the ellipse? Be careful when you differentiate the equation of the ellipse!

Do not forget that we know that the point \(P\) lies on the ellipse.