A point moves so that its distance from the point (2, 0) is always $\frac{1}{3}$ of its distance from the line x- 18 = 0. If the
locus of the point is a conic, its length of latus rectum is

 Clearly locus is ellipse with eccentricity $e=\frac{1}{3}$ 

Here focus is (2, 0) and directrix is x - 18 = 0.

 Distance of f:ocus from directrix $=16=a/e-ae$

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{8}{3}\mathrm{a}=16\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}=6\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{b}}^2={\mathrm{a}}^2\left(1-{\mathrm{e}}^2\right)=36(1-1/9)=32\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{L}\cdot \mathrm{R},=\frac{2{\mathrm{b}}^2}{\mathrm{a}}=\frac{32}{3}\end{array}$