Answer :

__Given:__ Equation of line is

_{To find:}_{image of point (0, 2, 3)}

__Formula Used:__ Equation of a line is

Vector form:

Cartesian form:

where is a point on the line and with b_{1} : b_{2} : b_{3} being the direction ratios of the line.

If 2 lines of direction ratios a_{1}:a_{2}:a_{3} and b_{1}:b_{2}:b_{3} are perpendicular, then a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3} = 0

_{Mid-point of line segment joining (}x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is

_{Explanation:}

Let

So the foot of the perpendicular is (5λ – 3, 2λ + 1, 3λ – 4)

The direction ratios of the perpendicular is

(5λ – 3 - 0) : (2λ + 1 - 2) : (3λ - 4 - 3)

⇒ (5λ – 3) : (2λ – 1) : (3λ – 7)

Direction ratio of the line is 5 : 2 : 3

From the direction ratio of the line and the direction ratio of its perpendicular, we have

5(5λ - 3) + 2(2λ – 1) + 3(3λ – 7) = 0

⇒ 25λ – 15 + 4λ – 2 + 9λ – 21 = 0

⇒ 38λ = 38

⇒ λ = 1

So, the foot of the perpendicular is (2, 3, -1)

The foot of the perpendicular is the mid-point of the line joining (0, 2, 3) and (α, β, γ)

So, we have

So, the image is (4, 4, -5)

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