If the directions cosines of a line are A, k, k, then

The correct option is k = 1/√3 or – (1/√3).

Step-by-step Explanation:

Let the direction cosines of the given line be represented as
$l$
,
$m$
, and
$n$
. Here, all three direction cosines are equal to
$k$
(representing the values of
$l$
,
$m$
, and
$n$
).

A fundamental identity for the direction cosines of any straight line in three-dimensional space states that the sum of their squares is equal to 1:

$$l^2+m^2+n^2=1$$

Substituting
$l=k$
,
$m=k$
, and
$n=k$
into this identity, we get:

$$k^2+k^2+k^2=1$$

Combining the terms on the left-hand side:

$$3k^2=1$$

Dividing both sides by 3 to isolate
$k^2$
:

$$k^2=\frac{1}{3}$$

Taking the square root of both sides gives:

$$k=\pm \frac{1}{\sqrt{3}}$$

This results in the two possible values for
$k$
:

$$k=\frac{1}{\sqrt{3}}\text{ or }k=-\frac{1}{\sqrt{3}}$$