Let y = mx + c, m > 0 be the focal chord of y ^ 2 = - 64x , which...

We have,

${y}^{2} = - 64x$

Coordinates of focus of parabola is : \left(- 16, 0\right)$</p><p>$y = mx + c$is focal chord, hence$

$⇒ c = 16m . . . \left(1\right)$

$y = mx + c$ is tangent to $( x + 10{)}^{2} + {y}^{2} = 4$, therefore$</p><p>$y = m \left(x + 10\right)$\pm 2\sqrt{1 + {m}^{2}}$

$⇒ c = 10m \pm 2\sqrt{1 + {m}^{2}}$

$⇒ 16m = 10m \pm 2\sqrt{1 + {m}^{2}}$

$⇒ 6m = 2\sqrt{1 + {m}^{2}}, \left(m > 0\right)$

$⇒ 9{m}^{2} = 1 + {m}^{2}$

$⇒ m = \dfrac{1}{2\sqrt{2}} \text{\text{ and } } c = \dfrac{8}{\sqrt{2}}$

Then,$</p><p>$4\sqrt{2}\left(m + c\right) = 4\sqrt{2}\left(\dfrac{17}{2\sqrt{2}}\right) = 34