Find the equation of the plane through the line of intersection of the planes and and which is perpendicular to the plane ?

We know that, the equation of a plane through the line of intersection of the planes

and

is given by

So, equation of the plane passing through the line of intersection of the plane

and is given by

[] + k[] = 0

.[ + k[] – 4 + 5k = 0 …… (1)

We know that two planes perpendicular if

.

Given that plane (1) is perpendicular to the plane

Using (1)and (3) in equation (2),

[ + k()]( = 0

(1 + 2k)(5) + (2 + k)(3) + (3 – k)( – 6) = 0

5 + 10k + 6 + 3k – 18 + 6k = 0

19k – 7 = 0

k =

Put the value of k in equation (1),

.[)] – 4 + 5() = 0

.[] –

.[] –

Multiplying by 19,

Equation of required plane is,

33x + 45y + 50z – 41 = 0