Question

Let a, b, c be the direction ratios of a line then direction cosines are-

$\frac{a}{\sqrt{\sum a^{2}}}, \frac{b}{\sqrt{\sum a^{2}}}, \frac{c}{\sqrt{\sum a^{2}}}$
$\frac{1}{\sqrt{\sum a^{2}}}, \frac{1}{\sum a^{2}}, \frac{1}{\sum a^{2}}$
$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$
$\frac{a}{\sqrt{\sum a^{2}}}, \frac{b}{\sqrt{\sum b^{2}}}, \frac{c}{\sqrt{\sum c^{2}}}$

Answer: $\frac{a}{\sqrt{\sum a^{2}}}, \frac{b}{\sqrt{\sum a^{2}}}, \frac{c}{\sqrt{\sum a^{2}}}$

If $l, m, n$ are the direction cosines and $a, b, c$ are the direction ratios of a line then

l = \frac{a}{\sqrt{a^{2} + b^{2} + c^{2}}}; m = \frac{b}{\sqrt{a^{2} + b^{2} + c^{2}}}; n = \frac{c}{\sqrt{a^{2} + b^{2} + c^{2}}}

If we observe above given direction cosines then we see that nomerator of these cosines are different but denominator is same.

Now, observe in given four options we see that only option 1 is different nomerator with same denominator.
Hence, option 1 is correct.

So, correct answer is option-1, $\to \frac{a}{\sqrt{\sum a^{2}}}, \frac{b}{\sqrt{\sum a^{2}}}, \frac{c}{\sqrt{\sum a^{2}}}$