Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the planes $3 x-2 y+4 z-7=0$ and $x+5 y-2 z+9=0$, be $\alpha x+\beta y+\gamma z+3=0$, then $\alpha+\beta+\gamma$ is equal to:
Select the correct option:
A
-23
B
-15
C
23
D
15
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Solution
Here is the accurate text from the image:
Equation of plane is 3x − 2y + 4z − 7 + λ(x + 5y − 2z + 9) = 0
(3 + λ)x + (5λ − 2)y + (4 − 2λ)z + 9λ − 7 = 0
passing through (1, 4, −3)
⇒ 3 + λ + 20λ − 8 − 12 + 6λ + 9λ − 7 = 0
⇒ λ = 2/3
⇒ equation of plane is
−11x − 4y − 8z + 3 = 0
⇒ α + β + γ = −23
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