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

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Find the equation of the plane passing through the line of intersection of the planes $x + 2 y + 3 z - 5 = 0$ and $3 x - 2 y - z + 1 = 0$ and cutting off equal intercepts on the x-axis and z-axis.

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Jack Discussed

Find the equation of the plane passing through the line of intersection of the planes

x + 2 y + 3 z - 5 = 0

and

3 x - 2 y - z + 1 = 0

and cutting off equal intercepts on the x-axis and z-axis.

11 mins ago

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11 mins ago

Text solutionVerified

Let the required plane be

(x + 2 y + 3 z - 5) + λ (3 x - 2 y - z + 1) = 0

....... (i)

Then,

(1 + 3 λ) x + (2 - 2 λ) y + (3 - λ) z - 5 + λ = 0

(1 + 3 λ) x + (2 - 2 λ) y + (3 - λ) z = 5 - λ

\Rightarrow \frac{( 1 + 3 λ )}{( 5 - λ )} x + \frac{( 2 - 2 λ )}{( 5 - λ )} y + \frac{( 3 - λ )}{( 5 - λ )} z = 1

\Rightarrow \frac{x}{( \frac{5 - λ}{1 + 3 λ} )} + \frac{y}{( \frac{5 - λ}{2 - 2 λ} )} + \frac{z}{( \frac{5 - λ}{3 - λ} )} = 1

Since the intercepts on the x-axis and z-axis are equal, we have

\frac{5 - λ}{1 + 3 λ} = \frac{5 - λ}{3 - λ} \Rightarrow 3 - λ = 1 + 3 λ

\Rightarrow 4 λ = 2 \Rightarrow λ = \frac{1}{2}

Putting in (i) we get equation of required plane.

5 x + 2 y + 5 z - 9 = 0

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Question Text	Find the equation of the plane passing through the line of intersection of the planes $x + 2 y + 3 z - 5 = 0$ and $3 x - 2 y - z + 1 = 0$ and cutting off equal intercepts on the x-axis and z-axis.
Answer Type	Text solution:1
Upvotes	150