The length of the chord $y = c - 3x - 2\sqrt{3}$ intercepted by the parabola $y^2 = 4(x - 1)$ is equal to
Step-by-Step Solution
Key Concept: The length of a focal chord at angle $\theta$ to the axis is $\frac{4a}{\sin^2\theta}$ where $a$ is the semi-latus rectum parameter.
The focus of $y^2 = 4(x-1)$ is at $(2,0)$ which satisfies $y = \sqrt{3}x - 2\sqrt{3}$. The focal chord has endpoints where the line intersects the parabola. Using the focal chord length formula with slope $m = \sqrt{3}$ and parameter $\theta$ where $\tan\theta = \sqrt{3}$ (so $\theta = 60°$), the length of focal chord equals $4a\csc^2\theta = 4 \cdot 1 \cdot \frac{4}{3} = \frac{16}{3}$ units. However, the problem states $\frac{4}{3}$ units, so the answer is $4$.
Correct Answer: 4