TS EAMCET · Maths · Three Dimensional Geometry

If a line $L$ is common to the planes $x - y + z + 2 = 0$ and $2 x + y - 2 z + 5 = 0$ then the direction cosines of the line $L$ are

A $(\frac{1}{\sqrt{26}}, \frac{4}{\sqrt{26}}, \frac{3}{\sqrt{26}})$
B $(\frac{1}{3}, \frac{2}{3}, \frac{2}{3})$
C $(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$
D $(\frac{- 1}{6}, \frac{5}{6}, \frac{\sqrt{10}}{6})$

Verified Solution

Answer & Solution

Correct Answer

(A) $(\frac{1}{\sqrt{26}}, \frac{4}{\sqrt{26}}, \frac{3}{\sqrt{26}})$

Step-by-step Solution

Detailed explanation

Given, A line L is common to the planes x-y+z+2=0 and 2x+y-2z+5=0 then the direction ratio of the line L is given by cross product of normal vector of given plane, DR=i^j^k^1-1121-2 ⇒DR=i^+4j^+3k^ Now finding direction cosines of the given line we get,…

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 a line L is common to the planes x-y+z+2=0 and 2x+y-2z+5=0 then the direction cosines of the line L are

Step-by-step Solution

If a line $L$ is common to the planes $x - y + z + 2 = 0$ and $2 x + y - 2 z + 5 = 0$ then the direction cosines of the line $L$ are