Question

If $L_1$ and $L_2$ are the length of the segments of any focal chord of the parabola $y^2=x$, then $\frac{1}{L_1}+\frac{1}{L_2}$ is equal to

• A. $2$
• B. $3$
• C. $4$
• D. none of these

Answer 1

The correct option is C $4$

Any point on the parabola is $P(a{t}^2,2at)$ where $a=\frac{1}{4}$.

$S(a,0)$ is the focus of the parabola. So,
$L_1=SP=\sqrt{(a{t}^2-1)+{\left(2at\right)}^2}=a(t^2+1)$

If $PQ$ is a focal chord, then $Q(\frac{a}{t^2},-\frac{2a}{t})$ (since for focal chord $t_1{t}_2=-1$)

So $L_2=SQ=a(\frac{1}{t^2}+1)=\frac{a(1+t^2)}{t^2}$
$\Rightarrow \frac{1}{L_1}+\frac{1}{L_2}=\frac{1}{a}=4$.