Inclined tangents

Let one extremity of a focal chord of the ellipse be at $(h, k)$ . Then the other extremity will be at $\left (\dfrac {75-17h}{17-3h},-\dfrac {8k}{17-3h}\right )$ .

Slopes of the tangents at these points are $-\dfrac {16h}{25k}$ and $\dfrac {2(75-17h)}{25k}$ .

The tangent of the angle between these two lines is

$\tan α=\dfrac{-\frac{16h}{25k}-\frac{150-34h}{25k}}{1-\frac{16h}{25k}\times \frac{150-34h}{25k}}$

$=\dfrac{75k}{8(3h-25)}$ .

Substituting the value of $k$ and simplifying we get the range of the ratio as $-\dfrac {15}{8}\leq \dfrac{75k}{8(3h-25)}\leq \dfrac{15}{8}$ .

Hence the required maximum value is $\dfrac{15}{8}=1.875\implies$ the maximum angle is $\tan^{-1} (1.875)\approx 61.927\degree$ .