Three Dimensional Geometry

Equation of required plane is
P ≡ (x + 2y + 3z - 2) + λ(x - y + z - 3) = 0
⇒ (1+ λ )x + (2 -λ ) y + (3 + λ )z - (2 + 3λ) = 0
Its distance from (3, 1, −1) is

2

√3

.
⇒

2

√3

=
=

4

3

=

(−2λ)2

2λ2+4λ+14

⇒ 3λ2+4λ+14 = 3λ2
⇒ λ = −

7

2

⇒ −

5

2

x+

11

2

y−

z

2

+

17

2

= 0
- 5x + 11 y - z + 17 = 0