For all d, 0 < d < 1, which one of the following points is the reflection of the point (d, 2d, 3d) in the plane passing through the points (1, 0, 0), (0, 1, 0) and (0, 0, 1) ?
Answer & explanation
Correct answer: option 1
The equation of the plane passing through (1, 0, 0), (0, 1, 0) and (0, 0, 1) is x + y + z = 1. Let $(\alpha, \beta, \gamma )$ be the reflection of the point (d, 2d, 3d) in the plane x + y + z = 1. Then,
$\frac{\alpha - d}{1}=\frac{\beta - 2d}{1}=\frac{\gamma - 3d}{1}= -\frac{2(d+2d+3d-1)}{1^2+1^2+1^2}$
$⇒ \alpha - d = \beta - 2d = \gamma - 3d = -\frac{2}{3} (6d-1)$
$⇒\alpha - d = \beta - 2d = \gamma - 3d = -4d + \frac{2}{3}$
$⇒\alpha = \frac{2}{3} - 3d, \beta = \frac{2}{3} - 2d, \gamma = \frac{2}{3} - d$