Question

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

Answer 1

Since the line lies on the given plane, therefore the point $\left(a , 2 d , c\right)$ 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, $b^{2}=2bd\Rightarrow \frac{b}{d}=2$ .