If the mirror image of the point P(3,4,9) in the line x−13=y+12=z−21 is (α,β,γ), then 14(α+β+γ) is :

The mirror image of a point in a line is given by the formula:

P'(x', y', z') = P(x, y, z) - 2 * [(x - x1)(a) + (y - y1)(b) + (z - z1)(c)] / (a^2 + b^2 + c^2) * (a, b, c)

Where P(x, y, z) is the point, P'(x', y', z') is the mirror image, (x1, y1, z1) is a point on the line, and (a, b, c) are the direction ratios of the line.

Given the line x−13=y+12=z−21, we can see that (x1, y1, z1) = (13, -12, 21) and (a, b, c) = (1, 1, 1).

Given the point P(3, 4, 9), we can substitute these values into the formula to find the mirror image P'(α, β, γ):

P'(α, β, γ) = P(3, 4, 9) - 2 * [(3 - 13)(1) + (4 - (-12))(1) + (9 - 21)(1)] / (1^2 + 1^2 + 1^2) * (1, 1, 1) = (3, 4, 9) - 2 * [-10 + 16 - 12] / 3 * (1, 1, 1) = (3, 4, 9) - 2 * [-6] / 3 * (1, 1, 1) = (3, 4, 9) + 4 * (1, 1, 1) = (7, 8, 13)

So, α = 7, β = 8, γ = 13.

Finally, substituting these values into the expression 14(α+β+γ), we get:

14(α+β+γ) = 14(7 + 8 + 13) = 14 * 28 = 392.