Question

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 :

• A. $\frac{16}{3}$
• B. $\frac{32}{3}$
• C. $\frac{8}{3}$
• D. $\frac{15}{4}$

Answer 1

Answer: (B)

$\frac{32}{3}$

We know that if ratio of distance of point from fixed point and from fixed line is constant(<1),then locus of such point is ellipse and constant is eccentricity.
From given data,$e=\frac{1}{3}$ and fixed line is $x-18=0$,its distance from origin is $\frac{a}{e}=18⟹a=6$,from definition of eccentricity ,we get value of $b^2=32$, where $2a$ and $2b$ are lengths of major and minor axis respectively.
we know that length of latus rectum is $\frac{2b^2}{a}=\frac{32}{3}$.