Problem

The hyperbola H has equation

$$\frac{x^2}{64} - \frac{y^2}{49} = 1$$

and eccentricity $e$.

(a) Show that

$$e = \frac{113}{8}$$

The point $(8\sec\theta, 7\tan\theta)$, where $0<\theta<\frac{\pi}{2}$, lies on H.

(b) Use calculus to show that the tangent to H at P has equation

$$\frac{x}{8}\sec\theta - \frac{y}{7}\tan\theta = 1$$

The tangent to H at P meets the y-axis at the point Q.

(c) Write down the coordinates of Q.

The normal to H at P

• has equation $8x\cos\theta + 7y\cot\theta = 113$
• meets the y-axis at the point R

(d) Write down the coordinates of R.

(e) Hence show that the circle with QR as a diameter passes through the foci of H.