Q.If the direction cosines of a line are $k, k, k$, then
(A) $k > 0$
(B) $0 < k < 1$
(C) $k = 1$
(D) $k = \dfrac{1}{\sqrt{3}}$ or $-\dfrac{1}{\sqrt{3}}$

Concept: Direction Cosines Property — For any line, the sum of squares of its direction cosines equals 1.

Given direction cosines are $k, k, k$. Using the property:

This simplifies to:

So: …

The key idea here is that direction cosines are not just any three numbers — they are the cosines of the angles a line makes with the coordinate axes. Because of that geometric meaning, they must satisfy a specific constraint.

Why the sum-of-squares condition?

If a line makes angles $\alpha, \beta, \gamma$ with the $x, y, z$ axes respectively, then its direction cosines are $l = \cos\alpha$, $m = \cos\beta$, $n = \cos\gamma$. A fundamental identity from 3D geometry states that for any line,

This is not arbitrary — it follows from the fact that the unit vector along the line has components $(\cos\alpha, \cos\beta, \cos\gamma)$, and its magnitude must be $1$.

So whenever you see "direction cosines are given as something", the first thing to check is whether they satisfy $l^2 + m^2 + n^2 = 1$.

Now let's apply this to the problem.

1. Set up the condition.

   We are told the direction cosines are $k, k, k$. That means $l = k$, $m = k$, $n = k$.

2. Apply the fundamental identity.

3. Solve for $k$. $3k^2 = 1 \quad\Rightarrow\quad k^2 = \frac{1}{3} \quad\Rightarrow\quad k = \pm \frac{1}{\sqrt{3}}.$ …

Method: Using the Direction-Cosine Identity $l^2+m^2+n^2=1$

Whenever direction cosines are given with unknowns, the single governing fact is that their squares add to one.

Steps

Step 1: Recall the identity.

For direction cosines $l,m,n$ (the cosines of the angles a line makes with the axes),

It is the squares that sum to $1$ — never $l+m+n=1$.

Step 2: Substitute the given expressions.

Replace $l,m,n$ with the given values (here each equals $k$) and form the equation, e.g. $k^2+k^2+k^2=1$.

Step 3: Solve, and keep both signs. …

Common Mistakes

Mistake 1: Using $l+m+n=1$ instead of $l^2+m^2+n^2=1$.

Why it's wrong: the identity is about the squares summing to one; the plain sum has no such rule. Correct approach: square the direction cosines before adding, giving $3k^2=1$.

Mistake 2: Discarding the negative root. …