A point moves so that its distance from the point (2,0) is always (1)/(3)` of its distances from the line x-18. If the locus of the points is a conic, then length of its latus rectum is

Options

A. `(16)/(3)`
B. `(32)/(3)`
C. `(8)/(3)`
D. `(15)/(4)`

Explanation

To solve the problem step by step, we need to find the locus of a point that maintains a specific ratio of distances from a point and a line. Let's break it down:

### Step 1: Understand the given information
We have a point \( P(x, y) \) that moves such that its distance from the point \( (2, 0) \) is \( \frac{1}{3} \) of its distance from the line \( x - 18 = 0 \) (which is the vertical line \( x = 18 \)).

### Step 2: Write the distance equations
1. The distance from the point \( P(x, y) \) to the point \( (2, 0) \) is given by:
\[
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