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 $4a$ $cosec^{2}\theta $ where $a=1$ and $tan \theta =\sqrt{3 }\left(o r \theta = 6 0^{o}\right)$
$\Rightarrow $ length of chord $=4\times \left(c o s e c^{2} \left(60\right)^{o}\right)$
$=4\times \frac{4}{3}=\frac{16}{3}$ units