If the x-intercept of a focal chord of the parabola $y^2 = 8x + 4y + 4$ is 3, then the length of this chord is equal to ___.
Step-by-Step Solution
Key Concept: Convert to standard form, identify focus, use the focal chord through a point on x-axis formula
$y^2 - 4y = 8x+4 \Rightarrow (y-2)^2 = 8(x+1)$. Standard form with $X=x+1$, $Y=y-2$, $a=2$. Focus at $(1,2)$ in original. Focal chord passes through $(3,0)$. Slope: $m = (0-2)/(3-1) = -1$. Line through focus: $Y-0 = -1(X-2) \Rightarrow Y = -X+2$. Using focal chord length formula $\ell = 4a/\sin^2\theta$ where $\tan\theta = 1$: $\ell = 4\times 2/(1/2) = 16$. Answer: 16
Correct Answer: 16