Answer :

Given: Equation of line is


To find: image of point (5, 9, 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 (2λ + 1, 3λ + 2, 4λ + 3)


The direction ratios of the perpendicular is


(2λ + 1 - 5) : (3λ + 2 - 9) : (4λ + 3 - 3)


(2λ – 4) : (3λ – 7) : (4λ)


Direction ratio of the line is 2 : 3 : 4



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


2(2λ – 4) + 3(3λ – 7) + 4(4λ) = 0


4λ – 8 + 9λ – 21 + 16λ = 0


29λ = 29


λ = 1


So, the foot of the perpendicular is (3, 5, 7)


The foot of the perpendicular is the mid-point of the line joining (5, 9, 3) and (α, β, γ)


So, we have





So, the image is (1, 1, 11)

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