Find the direction ratios of the normal to the plane passing through the point (2,1, 3) and the line of intersection of the planes x + 2 y + z = 3 and 2 x – y – z = 5.

Solution

The equation of any plane through the intersection of planes
x + 2y + z – 3 = 0 and 2x – y – z – 5 = 0 is
(x + 2y + z – 3) + k (2x – y – z – 5) = 0 ...(1)
∴ it passes through (2, 1, 3)
∴ (2 + 2 + 3 – 3) + k (4 – 1– 3 – 5) = 0
$therefore space space space 4 space minus space 5 space straight k space equals space 0 space space space space space space space space space space space space rightwards double arrow space space space space straight k space equals space 4 over 5$
Putting $straight k space equals 4 over 5 space in space left parenthesis 1 right parenthesis comma space we space get comma$
$left parenthesis straight x plus 2 straight y plus straight z minus 3 right parenthesis plus space 4 over 5 left parenthesis 2 straight x minus straight y minus straight z minus 3 right parenthesis space equals space 0$
or 5 (x + 2 y + z – 3) + 4 (2 x – y – z – 3) = 0
or 5 x + 10 y + 5 z – 15 + 8 x – 4 y – 4 z – 12 = 0
or 13 x + 6 y + z – 27 = 0
which is equation of plane.
Direction ratios of normal to the plane are 13, 6, 1.

For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the direction ratios of the normal to the plane passing through the point (2,1, 3) and the line of intersection of the planes x + 2 y + z = 3 and 2 x – y – z = 5.

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Location

For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the direction ratios of the normal to the plane passing through the point (2,1, 3) and the line of intersection of the planes x + 2 y + z = 3 and 2 x – y – z = 5.

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin2α + sin2 β + sin2γ = 2.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.