Step 1: Understanding the Concept:
A tangent to a parabola passing through an external point must satisfy the quadratic equation for slopes. We use the sum and product of roots for these slopes. Step 2: Detailed Explanation:
1. Parabola: \(y^{2} = 64x \implies 4a = 64 \implies a = 16\).
2. Equation of tangent in slope form: \(y = mx + a/m \implies y = mx + 16/m\).
3. Since it passes through \((h, k)\):
\[ k = mh + 16/m \implies m^{2}h - mk + 16 = 0 \]
Let the slopes be \(m_{1}\) and \(m_{2}\). Given \(m_{1} = 8m_{2}\).
4. Sum of roots: \(m_{1} + m_{2} = 9m_{2} = \frac{k}{h} \implies m_{2} = \frac{k}{9h}\).
5. Product of roots: \(m_{1}m_{2} = 8m_{2}^{2} = \frac{16}{h} \implies m_{2}^{2} = \frac{2}{h}\).
6. Equate:
\[ \left( \frac{k}{9h} \right)^{2} = \frac{2}{h} \]
\[ \frac{k^{2}}{81h^{2}} = \frac{2}{h} \implies \frac{k^{2}}{h} = 162 \implies \frac{k^{2}}{2h} = 81 \] Step 3: Final Answer:
The value is 81.