Question

If the plane $ax-by+cz=0$ contains the line $\frac{x-a}{a}=\frac{y-2d}{b}=\frac{z-c}{c}$ , then the value of $\frac{b}{d}$ is equal to $\left(b,d\ne 0\right)$

Answer 1

Answer: 2

Since the line lies on the given plane, therefore the point $\left(a,2d,c\right)$ on the line also lies on the plane
$\Rightarrow a^2-2bd+c^2=0$
Also, the line will be perpendicular to the normal vector of the plane
$\Rightarrow a^2-b^2+c^2=0$
Hence, $b^2=2bd\Rightarrow \frac{b}{d}=2$ .