Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The equations of the given planes are $5x-2y+3z=9$ and $15x-6y+9z=-5$.

The direction ratios of the normal to the first plane are $a_1=5,b_1=-2,c_1=3$.

The direction ratios of the normal to the second plane are $a_2=15,b_2=-6,c_2=9$.

Taking the ratio of the corresponding direction ratios, we get:

$\frac{a_1}{a_2}=\frac{5}{15}=\frac{1}{3}$

$\frac{b_1}{b_2}=\frac{-2}{-6}=\frac{1}{3}$

$\frac{c_1}{c_2}=\frac{3}{9}=\frac{1}{3}$

Since $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, the normal vectors of the two planes are parallel.

Also, the ratio of the constant terms is $\frac{d_1}{d_2}=\frac{9}{-5}\ne \frac{1}{3}$, which means the planes are not coincident.

Therefore, the given planes are parallel.

Answer: Parallel