Find the equation of the plane passing through the line of intersectio

1.	Find the equation of the plane passing through the line of intersection of the planes vector r.(2i - 7j + 4k) = 3 and 3x – 5y + 4z + 11 = 0, and the point (-2, 1, 3).
Answer» Given planes are vector r.(2i - 7j + 4k) = 3 2x – 7y + 4z – 3 = 0 and 3x – 5y + 4z + 11 = 0 Equation of a plane which passes through the line of intersection of the planes (2x – 7y + 4z – 3) + λ(3x – 5y + 4z + 11) = 0 … (1) This passes through the point (-2, 1, 3). (1) ⇒ (-4 – 7 + 12 – 3) + λ(-6 – 5 + 12 + 11) = 0 -2 + λ(12) = 0 ⇒ 12λ = 2 λ = 2/12 ⇒ λ = 1/6 The required equation is (1) ⇒ (2x – 7y + 4z – 3) + (1/6)(3x – 5y + 4z + 11) = 0 12x – 42y + 24z – 18 + 3x – 5y + 4z + 11 = 0 15x – 47y + 28z – 7 = 0

Find the equation of the plane passing through the line of intersection of the planes vector r.(2i - 7j + 4k) = 3 and 3x – 5y + 4z + 11 = 0, and the point (-2, 1, 3).

Answer»

Given planes are

vector r.(2i - 7j + 4k) = 3

2x – 7y + 4z – 3 = 0 and 3x – 5y + 4z + 11 = 0

Equation of a plane which passes through the line of intersection of the planes

(2x – 7y + 4z – 3) + λ(3x – 5y + 4z + 11) = 0 … (1)

This passes through the point (-2, 1, 3).

(1) ⇒ (-4 – 7 + 12 – 3) + λ(-6 – 5 + 12 + 11) = 0

-2 + λ(12) = 0 ⇒ 12λ = 2

λ = 2/12 ⇒ λ = 1/6

The required equation is

(1) ⇒ (2x – 7y + 4z – 3) + (1/6)(3x – 5y + 4z + 11) = 0

12x – 42y + 24z – 18 + 3x – 5y + 4z + 11 = 0

15x – 47y + 28z – 7 = 0