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(A) The perpendicular distance of a point $P(x, y, z)$ from the $X$-axis is given by $d_1 = \sqrt{y^2 + z^2}$.The perpendicular distance of the point

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$4x^2 - 9y^2 - 9z^2 = 0$

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(A) The perpendicular distance of a point $P(x, y, z)$ from the $X$-axis is given by $d_1 = \sqrt{y^2 + z^2}$.The perpendicular distance of the point $P(x, y, z)$ from the $YZ$-plane is given by $d_2 = |x|$.According to the problem, the ratio of these distances is $d_1 : d_2 = 2 : 3$.Therefore, $\frac{\sqrt{y^2 + z^2}}{|x|} = \frac{2}{3}$.Squaring both sides, we get $\frac{y^2 + z^2}{x^2} = \frac{4}{9}$.Cross-multiplying, we obtain $9(y^2 + z^2) = 4x^2$.Rearranging the terms, we get $4x^2 - 9y^2 - 9z^2 = 0$.

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

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