Question

Equation of the plane which passes through the point (-1, 3, 2) & is $\perp$ to each of the planes $p_1$ & $p_2$ is

• A. $2x+y+z+1=0$
• B. $3x+8y-2z-17=0$
• C. $2x+y-z+1=0$
• D. $x-2y+z+1=0$

Answer 1

The correct option is C $2x+y-z+1=0$

Normal of $p1=[2-31]$
Normal of $p2=[1-11]$
If planes are perpendiculars then dot products of their normals must be 0,
So consider normal of desired plane is $\left[abc\right]$
Hence,
$2a-3b+c=0$ (Dot product with normal of p1)
$a-b+c=0$ (Dot product with normal of p2)
Solving above 2 we get ,
$a-2b=0$
So if we use normals of planes in option b & d,
we notice that they don't satisfy the above equation
We observe that option a does not pass through (-1, 3, 2)
Since on substitution ,
$2.(-1)+3+2+1\ne 0$

The correct answer is $2x+y-z+1=0$