Q. 7

# Find the length o

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(x1,y1,z1) and B(x2,y2,z2) on a line, then its direction ratios are proportional to (x2 – x1,y2 – y1,z2 – z1)’

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.”

a1a2 + b1b2 + c1c2 = 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(x1,y1,z1) and B(x2,y2,z2) is given by

units

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses