If $a$ , $b$ , $c$ are the lengths of the intercepts of the plane passing through the intersection of the planes $2 x + y + 2 z =: 9$ , $4 x - 5 y - 4 z = 1$ and the point $(3, 2, 1)$ , on the coordinate axes, then ${(5 a + b + c)}^{2}$ is equal to

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Solution

Planes are
$2 x + y + 2 z = 9 \to Π_{1}, 4 x - 5 y - 4 z = 1 \to Π_{2}$
Plane which is intersection of above planes is
$Π = Π_{1} + λ Π_{2}$
$Π = (2 x + y + 2 z - 9) + λ (4 x - 5 y - 4 z - 1) = 0 (1)$
passes through $(3, 2, 1)$
Put $x = 3, y = 2, z = 1$ in $(2)$
$(2 \times 3 + 2 + 2 - 9) + λ (12 - 10 - 4 - 1) = 0$
$λ = \frac{1}{3}$
Put $λ = \frac{1}{3}$ in equation $(1)$
$Π \Rightarrow 5 x - y + z = 14$
Length of intercept on X-axis is $\frac{14}{5} = a$
Length of intercept on Y-axis is $- 14 = b$
Length of intercept on Z-axis is $14 = c$
$(5 a + b + c)^{2} = {(5 \times \frac{14}{5} - 14 + 14)}^{2} = 196 (5 a + b + c)^{2} = 196$

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