Question

Let $(\alpha, \beta, \gamma)$ be the mirror image of the point $(2, 3, 5)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$. Then $2\alpha + 3\beta + 4\gamma$ is equal to

• A. 32
• B. 33
• C. 31
• D. 34

Answer 1

Answer: (B)

33

Answer 2

Let $P(2, 3, 5)$ be the point and $R(\alpha, \beta, \gamma)$ its mirror image in the line

$\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}.$

Since $R$ is the mirror image of $P$, the line segment $PR$ is perpendicular to the direction ratios of the line $(2, 3, 4)$.

Therefore, $\overrightarrow{PR} \perp (2, 3, 4)$.

So, $\overrightarrow{PR} \cdot (2, 3, 4) = 0$.

Let $\overrightarrow{PR} = (\alpha - 2, \beta - 3, \gamma - 5)$.

Now,

$(\alpha - 2, \beta - 3, \gamma - 5) \cdot (2, 3, 4) = 0$

which gives:

$2(\alpha - 2) + 3(\beta - 3) + 4(\gamma - 5) = 0$ $\implies 2\alpha + 3\beta + 4\gamma = 4 + 9 + 20 = 33$

Thus, the answer is:

33