Q. 34.7( 3 Votes )

# Find the foot of the perpendicular drawn from the point A(1, 0, 3) to the joint of the points B(4, 7, 1) and C(3, 5, 3).

Answer :

Given: - Perpendicular from A(1, 0, 3) drawn at line joining points B(4, 7, 1) and C(3, 5, 3)

Let D be the foot of the perpendicular drawn from A(1, 0, 3) to line joining points B(4, 7, 1) and C(3, 5, 3).

Now let's find the equation of the line which is formed by joining points B(4, 7, 1) and C(3, 5, 3)

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 = - λ + 4, y = – 2λ + 7, z = 2λ + 1

Therefore, coordinates of D( – λ + 4, – 2λ + 7, 2λ + 1)

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

= ( – λ + 4 – 1), ( – 2λ + 7 – 0), (2λ – 2)

= ( – λ + 3), ( – 2λ + 7), (2λ – 2)

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

(hint: denominator terms of line equation)

= ( – 1, – 2,2)

Since 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.

⇒ – 1( – λ + 3) + ( – 2)( – 2λ + 7) + 2(2λ – 2) = 0

⇒ λ – 3 + 4λ – 14 + 4λ – 4 = 0

⇒ 9λ – 21 = 0

Therefore coordinates of D

i.e Foot of perpendicular

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

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