How many lines through the origin in make equal angles with the coordinate axis?

The correct option is 8.

To understand why there are 8 such lines (or directions of lines) through the origin that make equal angles with the coordinate axes, we can use the concept of direction cosines in three-dimensional space.

Let a line passing through the origin make angles $\alpha$, $\beta$, and $\gamma$ with the positive directions of the x-axis, y-axis, and z-axis respectively. The direction cosines of this line are given by:
$l=\cos \alpha$
$m=\cos \beta$
$n=\cos \gamma$

A fundamental property of direction cosines is that the sum of their squares is always equal to 1:
$l^2+m^2+n^2=1$
Substituting the cosine values, we get:
${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

Since the line makes equal angles with the coordinate axes, the magnitudes of these angles (or their cosines) must be equal. Therefore, we can write:
${\cos }^2\alpha ={\cos }^2\beta ={\cos }^2\gamma$

Substituting this back into our identity:
$3{\cos }^2\alpha =1$
This simplifies to:
${\cos }^2\alpha =\frac{1}{3}$
Taking the square root on both sides gives:
$\cos \alpha =\pm \frac{1}{\sqrt{3}}$

Since each of the three direction cosines can independently take a positive or negative value, we have:
$l=\pm \frac{1}{\sqrt{3}}$, $m=\pm \frac{1}{\sqrt{3}}$, and $n=\pm \frac{1}{\sqrt{3}}$

The total number of unique combinations for the signs of $\left(l,m,n\right)$ is given by:
$2^3=8$ combinations.

These 8 distinct sets of direction cosines correspond to the 8 directions through the origin that form equal angles with the coordinate axes:
1. $\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$
2. $\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right)$
3. $\left(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$
4. $\left(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right)$
5. $\left(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$
6. $\left(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right)$
7. $\left(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$
8. $\left(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right)$

Thus, considering these directional vectors or lines, there are 8 such lines/directions passing through the origin.