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

The Correct Option is B

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