TS EAMCET · Maths · Three Dimensional Geometry

The lines whose direction cosines are given by the relations $a l + b m + c n = 0$ and $m n + n l + l m = 0$ are

A perpendicular if $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$
B perpendicular if $\sqrt{a} + \sqrt{b} + \sqrt{c} = 0$
C parallel if $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$
D parallel if $a + b + c = 0$

Verified Solution

Answer & Solution

Correct Answer

(A) perpendicular if $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$

Step-by-step Solution

Detailed explanation

Given direction cosines are al+bm+cn=0 ...1 ⇒n=-al+bmc ...2 and mn+nl+lm=0 ...3 From equation 1, -mal+bmc-lal+bmc+lm=0 ⇒m(al+bm)+l(al+bm)-lcm=0 ⇒al2+lma+b-c+bm2=0 Dividing by m2, ⇒al2m2+lma+b-c+b=0 If l1, m1, n1…

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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The lines whose direction cosines are given by the relations al+bm+cn=0 and mn+nl+lm=0 are

Step-by-step Solution

The lines whose direction cosines are given by the relations $a l + b m + c n = 0$ and $m n + n l + l m = 0$ are