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  4. The reflection of the point A (1, 0, 0) in the line (x-1/2)=(y+1/-3)=-(z+10/8) is

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Q. The reflection of the point A (1, 0, 0) in the
line $\frac{x-1}{2}=\frac{y+1}{-3}=-\frac{z+10}{8}$ is

Introduction to Three Dimensional Geometry

Solution:

Any point P on the given line is $(2r + 1, - 3r -1, 8r- 10) $
So the direction ratios of AP are $2r, - 3r - 1, 8r - 10$.
Now AP is perpendicular to the given line if $2(2r) - 3(- 3r - 1) + 8 (8r - 10)$ = 0
$\Rightarrow $ $77r - 77 = 0 \, \Rightarrow \, r = 1 $
and thus the coordinates of P, the foot of the perpendicular from A on the line are $(3, -4, -2)$.
Let B $(a, b, c)$ be the reflection of A in the given line. Then P is the mid-point of AB
$\frac{a + 1}{2} = 3, \frac{b}{2} = - 4 , \frac{c}{2} = - 2$
$\Rightarrow $ $a$ = 5, $b$ = 8, $c$ = - 4
Thus the coordinates of required point are (5, -8, -4).