Question

show that the lines whose dc's are given by l + m + n = 0, 2mn + 3nl -5lm = 0 are perpendicular

Answer 1

The lines whose direction cosines are given by l + m + n = 0, 2mn + 3nl - 5lm = 0 are perpendicular.

Step-by-step explanation:

The direction cosines are connected by

l + m + n = 0 ..... (1)

2mn + 3nl - 5lm = 0 ..... (2)

From (1), we get

l = - m - n

Substituting l = - m - n in (2), we get

2mn + 3n (- m - n) - 5 (- m - n) m = 0

or, 2mn - 3mn - 3n² + 5m² + 5mn = 0

or, 5m² + 4mn - 3n² = 0

or, 5 (m/n)² + 4 (m/n) - 3 = 0

If m₁/n₁, m₂/n₂ be the roots of the above equation, then

m₁m₂ / n₁n₂ = - 3/5

or, m₁m₂/3 = n₁n₂/(- 5) ..... (3)

Again from (1), we get

m = - l - n

Substituting m = - l - n in (2), we get

2 (- l - n) n + 3nl - 5l (- l - n) = 0

or, - 2nl - 2n² + 3nl + 5l² + 5nl = 0

or, 2n² - 6nl - 5l² = 0

or, 2 (n/l)² - 6 (n/l) - 5 = 0

If n₁/l₁, n₂/l₂ be the roots of the above equation, then

n₁n₂ / l₁l₂ = - 5/2

or, n₁n₂/(- 5) = l₁l₂/2 ..... (4)

From (3) and (4), we get

l₁l₂/2 = m₁m₂/3 = n₁n₂/(- 5) = k (say).

∴ l₁l₂ + m₁m₂ + n₁n₂ = 2k + 3k - 5k = 0

Here (l₁, m₁, n₁) and (l₂, m₂, n₂) being the direction cosines of the two lines, we can conclude that the two given straight lines are perpendicular to each other. Thus, proved.

Reference link:

Find the direction cosines l, m, n of the two lines which are connected by the relation l + m + n = 0 and mn - 2nl - 2lm = 0. - https://brainly.in/question/12553093

Answer 2

Answer:

explain

5(m/n)^+4(m/n)=0

after this equation