Question

The length of the chord $y=\sqrt{3}x-2\sqrt{3}$ intercepted by the parabola $y^2=4\left(x-1\right)$ is equal to

• A. $4\sqrt{3}$ units
• B. $\frac{8}{3}$ units
• C. $\frac{16}{3}$ units
• D. $\frac{4}{\sqrt{3}}$ units

Answer 1

Answer: (C)

Focus of $y^2=4\left(x-1\right)$ is $\left(2,0\right)$ which satisfies the equation $y=\sqrt{3}x-2\sqrt{3}.$
Hence, line $y=\sqrt{3}x-2\sqrt{3}$ is a focal chord.
Now, the length of the focal chord equals to $4a$ $cose{c}^2\theta$ where $a=1$ and $tan\theta =\sqrt{3}\left(or\theta =60^o\right)$
$\Rightarrow$ length of chord $=4\times \left(cose{c}^2{\left(60\right)}^o\right)$
$=4\times \frac{4}{3}=\frac{16}{3}$ units