First slide

Planes in 3D

Question

The equation of the plane which is equally inclined to the lines $\frac{x - 1}{2} = \frac{y}{- 2} = \frac{z + 2}{- 1}$ and $\frac{x + 3}{8} = \frac{y - 4}{1} = \frac{z}{- 4}$ and passing through the origin is/are

Moderate

A

4x-5y-7z=0
B

2x+7y-z=0
C

3x-4y-z=0
D

x+2y-5z=0

Solution

The plane is equally inclined to the lines. Hence, it is perpendicular to the angle bisector of the vectors $2 \hat{i} - 2 \hat{j} - \hat{k} and 8 \hat{i} + \hat{j} - 4 \hat{k}$
Vector along the angle bisectors of the vectors are
$\begin{array}{l} \frac{2 \hat{i} - 2 \hat{j} - \hat{k}}{3} \pm \frac{8 \hat{i} + \hat{j} - 4 \hat{k}}{9} \\ or \frac{14 \hat{i} - 5 \hat{j} - 7 \hat{k}}{9} and \frac{- 2 \hat{i} - 7 \hat{j} + \hat{k}}{9} . \end{array}$
Hence, the equations of the planes are 14x - 5y - 7z =0 or 2x + 7y - z = 0

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