Question

A line OP where O $=$ $(0,0,0)$ makes equal angles with ox, oy, oz. The point on OP, which is at a distance of $6$ units from O is:

• A. $(\frac{6}{\sqrt{3}},\frac{6}{\sqrt{3}},\frac{6}{\sqrt{3}})$
• B. $(2\sqrt{3},-2\sqrt{3},2\sqrt{3})$
• C. $-(2\sqrt{3},2\sqrt{3},2\sqrt{3})$
• D. $(6\sqrt{3},6\sqrt{3},6\sqrt{3})$

Answer 1

The correct option is A

$(\frac{6}{\sqrt{3}},\frac{6}{\sqrt{3}},\frac{6}{\sqrt{3}})$

Let $l,m,n$ be the direction cosines.

We have $l^2+m^2+n^2=1$.

Given that $l=m=n$ $\Rightarrow 3l^2=1$.

Solving we get $l=\frac{1}{\sqrt{3}}$.

Any point on the line is a multiple of direction cosines.

Let the point P is given by $(\frac{k}{\sqrt{3}},\frac{k}{\sqrt{3}},\frac{k}{\sqrt{3}})$.

Using the distance formula we get, $k=6$.