Find the equation of the plane passing through the point (−1, − 1, 2) and perpendicular to each of the following planes: 2x + 3y – 3z = 2 and 5x – 4y + z = 6

Solution

The equation of the piane passing through the point ( -1, -1, 2 ) is:

a( x + 1 ) + b( y + 1 ) + c ( z - 1 )= 0 ............(1)

where a, b and c are the direction ratios of the normal to the plane

It is given that the plane (1) is perpendicularto the planes.

2x + 3y - 3z = 2 and 5x - 4y + z = 6

$∴$ 2a + 3b - 3c = 0 ..............(2)

5a - 4b + c = 0 ...............(3)

solving equations (2) and (3), we have:

$\frac{a}{(3 x 1) - (- 4 x (- 3))} = \frac{b}{(- 3 x 5) - (2 x 1)} = \frac{c}{(2 x (- 4)) - (3 x 5)} \Rightarrow \frac{a}{- 9} = \frac{b}{- 17} = \frac{c}{- 23}$

So the direction ratios of the normal to the required plane are multiples of 9,

17, and 23.

Thus, the equation of the required plane is:

9(x + 1) + 17( y + 1) + 23( z - 2) = 0

or 9x + 17y + 23z = 20

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Three Dimensional Geometry

Find the equation of the plane passing through the point (−1, − 1, 2) and perpendicular to each of the following planes: 2x + 3y – 3z = 2 and 5x – 4y + z = 6

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.

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For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the equation of the plane passing through the point (−1, − 1, 2) and perpendicular to each of the following planes: 2x + 3y – 3z = 2 and 5x – 4y + z = 6

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.