Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
+1 vote
1.2k views
in 3D Coordinate Geometry by (55.4k points)
closed by

Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.

1 Answer

+1 vote
by (50.2k points)
selected by
 
Best answer

Given,

2l + 2m – n = 0 [1]

⇒ n = 2(l + m) [2]

and

mn + nl + lm = 0

⇒ 2m(l + m) + 2(l + m)l + lm = 0

⇒ 2lm + 2m2 + 2l2 + 2lm + lm = 0

⇒ 2m2 + 5lm + 2l2 = 0

⇒ 2m2 + 4lm + lm + 2l2 = 0

⇒ (2m + l)(m + 2l) = 0

So, we have two cases,

l = -2m

⇒ -4m + 2m – n = 0 [From 1]

⇒ n = 2m

Hence, direction ratios of one line is proportional to -2m, m, -2m or direction ratios are (2, 1, -2)

Another case is,

m = -2l

⇒ 2l + 2(-2l) – n = 0

⇒ 2l – 4l = n

⇒ n = -2l

Hence, direction ratios of another lines is proportional to l, -2l, -2l or direction ratios are (1, -2, -2)

⇒ cos θ = 0

⇒ θ = 90°

Hence, angle between given two lines is 90°

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...