JEE Mathematics 2024 Question with Solution

Given: The point is $(2, 3, 5)$ and its mirror image in the line $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ is $({\alpha}, {\beta}, {\gamma})$.

Find: The value of $2\alpha + 3\beta + 4\gamma$.

The direction ratios of the given line are $(2, 3, 4)$.

Since $({\alpha}, {\beta}, {\gamma})$ is the mirror image of $(2, 3, 5)$ in the given line, the segment joining the point and its image is perpendicular to the line. Hence,

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

So,

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

Therefore, the correct option is B.

Given: The line is

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

and the point is $(2, 3, 5)$.

Find: The value of $2\alpha + 3\beta + 4\gamma$ where $({\alpha}, {\beta}, {\gamma})$ is the mirror image.

Write the line in parametric form:

$$x = 1 + 2t, \quad y = 2 + 3t, \quad z = 3 + 4t$$

Its direction vector is $(2, 3, 4)$.

For the reflected point, the vector from $(2, 3, 5)$ to $({\alpha}, {\beta}, {\gamma})$ must be perpendicular to the line, so

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

Expanding,

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

Thus, the value required is $33$.