First slide

Planes in 3D

Question

If the image of the point $P (1, - 2, 3)$ in the plane, $2 x + 3 y - 4 z + 22 = 0$ measured parallel to the line, $\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$ is Q, then PQ is equal to

Moderate

A

$2 \sqrt{42}$
B

$\sqrt{42}$
C

$6 \sqrt{5}$
D

$3 \sqrt{5}$

Solution

Equation of the line passing through $P (1, - 2, 3)$
with dr's (1,4,5) is $\frac{x - 1}{1} = \frac{y + 2}{4} = \frac{z - 3}{5} = t$
$R (t + 1, 4 t - 2, 5 t + 3)$ lies on the plane then t = 1 ,R (2,2,8)
$P Q = 2 P R = 2 \sqrt{1 + 16 + 25} = 2 \sqrt{42}$

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