The length of the chord y = 3 x - 2 3 intercepted by the parabola...

Focus of $y^{2} = 4 \left(x - 1\right) is \left(2,0\right) which satisfies$ the equationy = \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 $4acosec^{2}\theta$ where a = 1 and $tan \theta = \sqrt{3} \lleft(or\theta =6{0}^{o})$
$⇒$ length of chord $= 4 \times \left(cosec^{2}{60}^{o}\right)<br />$= 4 \times $\dfrac{4}{3} = \dfrac{16}{3} units$