Answer :

Given: - Perpendicular from A(1, 0, 4) drawn at line joining points B(0, – 11, 3) and C(2, – 3, 1)

and D be the foot of the perpendicular drawn from A(1, 0, 4) to line joining points B(0, – 11, 3) and C(2, – 3, 1).

Now let's find the equation of the line which is formed by joining points B(0, – 11, 3) and C(2, – 3, 1)

Tip: - Equation of a line joined by two points A(x_{1},y_{1},z_{1}) and B(x_{2},y_{2},z_{2}) is given by

Now

Therefore,

⇒ x = 2λ, y = 8λ – 11, z = – 2λ + 3

Therefore, coordinates of D(2λ, 8λ – 11, – 2λ + 3)

Now as we know (TIP) ‘if two points A(x_{1},y_{1},z_{1}) and B(x_{2},y_{2},z_{2}) on a line, then its direction ratios are proportional to (x_{2} – x_{1},y_{2} – y_{1},z_{2} – z_{1})’

Hence

Direction Ratios of AD

= (2λ – 1), (8λ – 11 – 0), ( – 2λ + 3 – 4)

= (2λ – 1), (8λ – 11), ( – 2λ – 1)

and by comparing with given line equation, direction ratios of the given line are

(hint: denominator terms of line equation)

= (2,8, – 2)

Since the AD is perpendicular to given line, therefore by “condition of perpendicularity.”

a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0 ; where a terms and b terms are direction ratio of lines which are perpendicular to each other.

⇒ 2(2λ – 1) + (8)(8λ – 11) – 2( – 2λ – 1) = 0

⇒ 4λ – 2 + 64λ – 88 + 4λ + 2 = 0

⇒ 72λ – 88 = 0

Therefore coordinates of D

i.e. Foot of perpendicular

By putting the value of λ in D coordinate equation, we get

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