First slide

Planes in 3D

Question

The plane passing through the point (-2, -2, 2) and containing the line joining the points (1, 1, 1) and (1, -1, 2) makes intercepts on the coordinate axes, the sum of whose lengths is

Moderate

A

3
B

4
C

6
D

12

Solution

Equation of the required plane is $a (x + 2) + b (y + 2) + c (z - 2) = 0$
$d . r .' s of the line passing through A (x_{1}, y_{1}, z_{1}), B (x_{2}, y_{2}, z_{2}) are (x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1})$
$∴d.r.'s of the line are < 0, 2, - 1 >$
$\Rightarrow 2 b - c = 0 and 3 a + b = 0$
$Put b = λ ∴ c = 2 λ; a = - \frac{λ}{3}$
$∴ - \frac{λ}{3} (x + 2) + λ (y + 2) + 2 λ (z - 2) = 0$
$\Rightarrow - x + 3 y + 6 z - 8 = 0$
$\Rightarrow \frac{x}{(- 8)} + \frac{y}{(\frac{8}{3})} + \frac{z}{(\frac{4}{3})} = 1$
$∴ s u m o f t h e l e n g t h s o f t h e i n t e r c e p t s = 8 + \frac{8}{3} + \frac{4}{3} = 12$

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