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Family of Planes Passing through the Intersection of Two Planes

If the equati...

Question

If the equation of the plane passing through the line of intersection of the planes $2 x - 7 y + 4 z - 3 = 0, 3 x - 5 y + 4 z + 11 = 0$ and the point $(- 2, 1, 3)$ is $a x + b y + c z - 7 = 0$ , then the value of $2 a + b + c - 7$ is

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Solution

Given :
$P_{1} : 2 x - 7 y + 4 z - 3 = 0 P_{2} : 3 x - 5 y + 4 z + 11 = 0$
Equation of the required plane can be written by using family of planes:
$P_{1} + λ P_{2} \Rightarrow (2 x - 7 y + 4 z - 3) + λ (3 x - 5 y + 4 z + 11) = 0$
It passes through $(- 2, 1, 3)$ , so
$(- 4 + 7 + 12 - 3) + λ (- 6 - 5 + 12 + 11) = 0 \Rightarrow - 2 + λ (12) = 0 \Rightarrow λ = \frac{1}{6}$
Therefore, the equation of plane is
$15 x - 47 y + 28 z - 7 = 0 ∴ 2 a + b + c - 7 = 30 - 47 + 28 - 7 = 4$

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