Question

Sol. [Key Concept: One root is twice the other ($\alpha,2\alpha$): use sum $= 3\alpha$ and product $= 2\alpha^{2}$ of the roots to find $k$, then the latus rectum of $y^{2} = 6kx$ is $6k$.]

$$\ \alpha + 2\alpha = \frac{- 9(k - 1)}{k^{2} - 15k + 27} = 3\alpha$$

$$\begin{array}{r} \alpha \times 2\alpha = \frac{18}{k^{2} - 15k + 27} = 2\alpha^{2}\#(2) \end{array}$$

solving (1) & (2)

$$k = 2 $$⇒ parabola will be $y^{2} = 6kx = 12x$

length of latus rectum $= 12$