The length of the chord of the parabola `y^2=x` which is bisected at the point (2, 1) is `2sqrt(3)` (b) `4sqrt(3)` (c) `3sqrt(2)` (d) `2sqrt(5)`

Options

A. `2sqrt(3)`
B. `4sqrt(3)`
C. `3sqrt(2)`
D. `2sqrt(5)`

Answer

D

Explanation

(4) Chord through (2,1) is
`(x-2)/(costheta)=(y-1)/(sintheta)=r` (1)
Solving (1) with the parabola `y^(2)=x`, we have
`(1+rsintheta)^(2)=2+rcostheta`
`orsin^(2)thetar^(2)+(2sintheta-costheta)r-1=0`
This equation has two roots : `r_(1)=ACandr_(2)=-BC`.
Then,
Sum of roots, `r_(1)+r_(2)=0`
`or2sintheta-costheta=0ortantheta=(1)/(2)`
`AB=|r_(1)-r_(2)|=sqrt((r_(1)+r_(2))^(2)-4r_(1)r_(2))`
`=sqrt(4(1)/(sin^(2)theta))=2sqrt(5)`