If the x-intercept of a focal chord of the parabola y^2=8x+4y+4 is 3, then the length of t…

Question

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 \_\_\_\_\_\_.

Answer: 16

Accepted Answer

$y^2-4y=8x+4\Rightarrow(y-2)^2=8(x+1)$. With $X=x+1,\ Y=y-2$: $Y^2=8X$ ($4a=8,\ a=2$); focus at $X=2\Rightarrow x=1,\ y=2$, i.e. focus $(1,2)$.

Line through the focus meeting $(3,0)$: slope $m=\dfrac{0-2}{3-1}=-1$, i.e. $\tan\theta=-1$, $\sin^2\theta=\tfrac12$.

$$\text{Length}=\frac{4a}{\sin^2\theta}=\frac{8}{1/2}=16.$$

Solutions · 1