Question

The equation of the plane passing through the line of intersection of the planes $2x-y=0$ and $3z-y=0$ and perpendicular to the plane $4x+5y-3z=8$ is:

• A. $2x+17y+9z=0$
• B. $28x-17y+9z=0$
• C. $2x+17y-9z=0$
• D. $28x-17y-9z=0$

Answer 1

The correct option is B $28x-17y+9z=0$

Given: $p_1:2x-y=0$ and
$p_2:3z-y=0$
Any plane through the line of intersection is
$p_1+\lambda p_2=0$
Any plane through the line of intersection is $(2x-y)+\lambda (3z-y)=0\cdots \left(i\right)$
The given plane is $4x+5y-3z-8=0\cdots \left(ii\right)$
Since $\left(i\right)$ and $\left(ii\right)$ are perpendicular to each other, we have:
$2\times 4-(1+\lambda )\times 5+3\lambda (-3)=0$
$\Rightarrow \lambda =\frac{3}{14}$
On substituting the value of $\lambda$ in equation $\left(i\right)$, we get $14(2x-y)+3(3z-y)=0$
$\Rightarrow 28x-17y+9z=0$