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 b1 : b2 : b3 being the direction ratios of the line.


If 2 lines of direction ratios a1:a2:a3 and b1:b2:b3 are perpendicular, then a1b1+a2b2+a3b3 = 0


Mid-point of line segment joining (x1, y1, z1) and (x2, y2, z2) 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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