1

Given that the distance of a variable point P(x, y) from the point (2, 0) is 2/3 times its distance from the line x = 9/2. We need to determine the locus of point P.

1. Equation of the line:
The equation of the line x = 9/2 is a vertical line passing through the point (9/2, y). Any point on this line will have an x-coordinate of 9/2.

2. Distance between the point and the line:
Let's consider a point P(x, y) on the line x = 9/2. The distance between P and the point (2, 0) can be calculated using the distance formula:

d1 = √[(x - 2)^2 + (y - 0)^2]

Similarly, the distance between P and the line x = 9/2 can be calculated as the absolute value of the difference between the x-coordinates of the two points:

d2 = |x - 9/2|

3. Relationship between the distances:
According to the given condition, the distance of P from (2, 0) is 2/3 times its distance from the line x = 9/2. Mathematically, this can be expressed as:

d1 = (2/3) * d2

4. Substituting the values:
Substituting the expressions for d1 and d2 from step 2 into the equation from step 3, we get:

√[(x - 2)^2 + y^2] = (2/3) * |x - 9/2|

Squaring both sides of the equation to remove the square root, we obtain:

(x - 2)^2 + y^2 = (4/9) * (x - 9/2)^2

Expanding and simplifying the equation, we get:

9(x - 2)^2 + 9y^2 = 4(x - 9/2)^2

9x^2 - 36x + 36 + 9y^2 = 4x^2 - 72x + 324

Simplifying further, we have:

5x^2 - 36y^2 + 108x - 288 = 0

5x^2 - 36y^2 + 108x = 288

Dividing the equation by 288, we get the standard form of an ellipse:

(x^2/57.6) - (y^2/8) = 1

Therefore, the locus of the variable point P is an ellipse.