Let the equation of the plane through the points $(- 1, 1, 1)$ and $(1, - 1, 1)$ and perpendicular to the plane $x + 2 y + 2 z = 7$ be $k x + m y - n z + p = 0$ . Find $k + m + n + p$ ?

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Solution

Any plane through $(- 1, 1, 1)$ is
$A (x + 1) + B (y - 1) + C (z - 1) = 0$ ...(1)
It passes through $(1, - 1, 1)$
$∴ 2 A - 2 B + 0 C = 0.$ ... $(2)$
It is perpendicular to the plane
$x + 2 y + 2 z = 7 ∴ A + 2 B + 2 C = 0.$ ... $(3)$
Solving $(2)$ and $(3),$ we get
$\frac{A}{- 4} = \frac{B}{- 4} = \frac{C}{6}$ or $\frac{A}{2} = \frac{B}{2} = \frac{C}{- 3} = a$ (say)
Thus, $A, B$ and $C$ are proportional to $2, 2$ and $- 3$ respectively
Substituting in $(1)$ , the equation of the plane is
$2 a (x + 1) + 2 a (y - 1) - 3 a (z - 1) = 0$
$\Rightarrow 2 x + 2 y - 3 z + 3 = 0$
$\Rightarrow k + m + n + p = 10$

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