Answer :

Given: - Point (5, 4, – 1) and equation of line

Let, PQ be the perpendicular drawn from P to given line whose endpoint/ foot is Q point.

As we know position vector is given by

Therefore,

Position vector of point P is

and, from a given line, we get

⇒

⇒

⇒

On comparing both sides we get,

⇒ x = 1 + 2λ, y = 9λ, z = 5λ

⇒ ; Equation of line

Thus, coordinates of Q i.e. General point on the given line

⇒ Q((1 + 2λ), 9λ, 5λ)

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 ratio of PQ is

= (2λ + 1 – 5), (9λ – 4), (5λ + 1)

= (2λ – 4), (9λ – 4), (5λ + 1)

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

(hint: denominator terms of line equation)

= (2,9,5)

Since PQ 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 – 4) + (9)(9λ – 4) + 5(5λ + 1) = 0

⇒ 4λ – 8 + 81λ – 36 + 25λ + 5 = 0

⇒ 110λ – 39 = 0

Therefore coordinates of Q

i.e. Foot of perpendicular

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

Now,

Distance between PQ

Tip: - Distance between two points A(x_{1},y_{1},z_{1}) and B(x_{2},y_{2},z_{2}) is given by

units

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