First slide

Planes in 3D

Question

$If the reflection of the point P (1, 0, 0) in the line \frac{x - 1}{2} = \frac{y + 1}{- 3} = \frac{z + 10}{8} is (a, b, c) then$ $\frac{| a + b + c |}{7} =$

Moderate

A

1
B

3
C

4
D

7

Solution

Let M be the midpoint of PQ and AM=r
$\Rightarrow M (1 + 2 r, - 1 - 3 r, - 10 + 8 r)$
$\begin{array}{l} Since \bar{AM} ⊥ \bar{PM}, we have \bar{AM} . \bar{PM} = 0 \\ \Rightarrow r = 1 \\ \Rightarrow M (3, - 4, - 2) \\ ∴ Q = (a, b, c) = 2 M - P = (5, - 8, - 4) \\ ∴ \frac{| a + b + c |}{7} = \frac{| 5 - 8 - 4 |}{7} = \frac{7}{7} = 1 \end{array}$

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