Question

If the lengths of each of the segments of a focal chord of the parabola $y^2=8x,$ which are divided by the focus of the parabola are integers, then the length of the focal chord may be

• A. $8$
• B. $9$
• C. $12$
• D. $15$

Answer 1

Answer: (A), (B)

$9$

Let lengths of segments of focal chord be $m$ and $n.$
We know that,
$\frac{1}{m}+\frac{1}{n}=\frac{1}{a}\Rightarrow \frac{1}{m}+\frac{1}{n}=\frac{1}{2}\Rightarrow (m-2)(n-2)=4$
$\Rightarrow$ These factors are $4$ and $1$ or $2$ and $2$
For $4\times 1,m=6,n=3$ or $m=3,n=6$
$\Rightarrow m+n=6+3=9$

For $2\times 2,m=4,n=4$
$\Rightarrow m+n=4+4=8$