Question

If the ratio of the perpendicular distances of a variable point \( P(x,y,z) \) from the X-axis and from the YZ-plane is 2:3, then the equation of the locus of \( P \) is:

• A. \( 4x^2 - 9y^2 - 9z^2 = 0 \)
• B. \( 9x^2 - 4y^2 - 4z^2 = 0 \)
• C. \( 4x^2 - 4y^2 - 9z^2 = 0 \)
• D. \( 9x^2 - 9y^2 - 4z^2 = 0 \)

Hint:

For locus problems involving distance ratios, express distances explicitly and use algebraic manipulations.

Answer 1

The Correct Option is A

Step 1: Express distances in terms of coordinates.
The perpendicular distance from a point \( P(x, y, z) \) to the X-axis is: \[ d_1 = \sqrt{y^2 + z^2} \] The perpendicular distance from \( P \) to the YZ-plane is: \[ d_2 = |x| \] Step 2: Set up the ratio condition.
\[ \frac{d_1}{d_2} = \frac{2}{3} \] \[ \frac{\sqrt{y^2 + z^2}}{|x|} = \frac{2}{3} \] Step 3: Squaring both sides.
\[ 4(x^2) = 9(y^2 + z^2) \] \[ 4x^2 - 9y^2 - 9z^2 = 0 \] Thus, the required equation is: \[ \mathbf{4x^2 - 9y^2 - 9z^2 = 0} \]