TS EAMCET · Maths · Three Dimensional Geometry

If $l, m, n$ are the direction cosines of a line which makes angles $α, β$ and $γ$ with the coordinate axes $X, Y, Z$ , respectively, then $l m + m n + n l$ takes the maximum value when

A $α, β, γ$ are in arithmetic progression
B $α = β = γ$
C any two of $α, β, γ$ are the same
D one of $α, β, γ$ is zero and the remaining two are non-zero and unequal.

Verified Solution

Answer & Solution

Correct Answer

(B) $α = β = γ$

Step-by-step Solution

Detailed explanation

As we know, l=cosα m=cosβ n=cosγ Also, cos2α+cos2β+cos2γ=1 Now lm+mn+nl=cosαcosβ+cosβcosγ+cosγcosα lm+mn+nl will take the maximum value only when α=β=γ, that will be lm+mn+nl=3cos2α=1

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If l, m, n are the direction cosines of a line which makes angles α, β and γ with the coordinate axes X, Y, Z, respectively, then lm+mn+nl takes the maximum value when

Step-by-step Solution

If $l, m, n$ are the direction cosines of a line which makes angles $α, β$ and $γ$ with the coordinate axes $X, Y, Z$ , respectively, then $l m + m n + n l$ takes the maximum value when