Question:

**Step 1: Identify the parabola parameters.** The given parabola is $y^2 = 8x$. Comparing this with the standard form of a parabola $y^2 = 4ax$, we find $4a = 8$, which implies $a = 2$. The focus of the parabola is at $F(a, 0)$, so the focus is $F(2, 0)$. **Step 2: Verify that (8, 8) lies on the parabola.** Substitute the coordinates $(8, 8)$ into the parabola equation $y^2 = 8x$: $8^2 = 64$ $8(8) = 64$ Since $64 = 64$, the point $(8, 8)$ lies on the parabola. **Step 3: Determine the coordinates of the second endpoint of the focal chord.** Let the two endpoints of the focal chord be $(x_1, y_1)$ and $(x_2, y_2)$. We are given $(x_1, y_1) = (8, 8)$. For a focal chord of the parabola $y^2 = 4ax$, the product of the y-coordinates of its endpoints is $y_1y_2 = -4a^2$. Using $y_1 = 8$ and $a = 2$: $$8 \cdot y_2 = -4(2)^2$$ $$8y_2 = -4(4)$$ $$8y_2 = -16$$ $$y_2 = -2$$ **Step 4: Find the x-coordinate of the second endpoint.** Since the second endpoint $(x_2, y_2)$ lies on the parabola $y^2 = 8x$: $$(-2)^2 = 8x_2$$ $$4 = 8x_2$$ $$x_2 = \frac{4}{8} = \frac{1}{2}$$ Thus, the second endpoint of the focal chord is $\left(\frac{1}{2}, -2\right)$. **Step 5: Calculate the length of the focal chord.** The length of the focal chord is the distance between the two endpoints $(x_1, y_1) = (8, 8)$ and $(x_2, y_2) = \left(\frac{1}{2}, -2\right)$. Using the distance formula: $$L = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$ $$L = \sqrt{\left(8 - \frac{1}{2}\right)^2 + (8 - (-2))^2}$$ $$L = \sqrt{\left(\frac{16 - 1}{2}\right)^2 + (8 + 2)^2}$$ $$L = \sqrt{\left(\frac{15}{2}\right)^2 + (10)^2}$$ $$L = \sqrt{\frac{225}{4} + 100}$$ $$L = \sqrt{\frac{225}{4} + \frac{400}{4}}$$ $$L = \sqrt{\frac{625}{4}}$$ $$L = \frac{\sqrt{625}}{\sqrt{4}}$$ $$L = \frac{25}{2}$$ The length of the focal chord is $\frac{25}{2}$.