Question

The length of the chord of the parabola $y^2=x$ which is bisected at the point $(2,1)$ is

• A. $2\sqrt{3}$
• B. $4\sqrt{3}$
• C. $3\sqrt{2}$
• D. $2\sqrt{5}$

Answer 1

The correct option is D $2\sqrt{5}$

Chord through $(2,1)$ is $\frac{x-2}{\cos \theta }=\frac{y-1}{\sin \theta }=r$ ... (i)
Solving equation (i) with parabola $y^2=x$, we have
$(1+r\sin \theta {)}^2=2+r\cos \theta$
$\Rightarrow {\sin }^2\theta r^2+(2\sin \theta -\cos \theta )r-1=0$
This equation has two roots $r_1=AC$ and $r_2=-BC$
Then, sum of roots $r_1+r_2=0$
$\Rightarrow 2\sin \theta -\cos \theta =0\Rightarrow \tan \theta =\frac{1}{2}$
$AB=|r_1-r_2|$
$=\sqrt{(r_1+r_2{)}^2-4r_1{r}_2}$
$=\sqrt{4\frac{1}{{\sin }^2\theta }}$

$=2\sqrt{5}$