Question

If a straight line is equally inclined at an angle $ \theta $ with all the three coordinate axes, then $ \tan \theta =$

• A. \( 2\sqrt{2} \)
• B. \( \sqrt{2} \)
• C. \( 1 \)
• D. \( 1 + \sqrt{5} \)

Hint:

For lines equally inclined with the coordinate axes, use the fact that the sum of the squares of the direction cosines equals 1. This helps in finding the tangent of the angle of inclination.

Answer 1

The Correct Option is B

Step 1: Let the direction cosines of the line be \( \cos \theta \), which is the same for each axis since the line is equally inclined with all three axes.
Step 2: Using the fact that the sum of the squares of the direction cosines is equal to 1 for any line in three-dimensional space: \[ \cos^2 \theta + \cos^2 \theta + \cos^2 \theta = 1 \] This simplifies to: \[ 3 \cos^2 \theta = 1 \] \[ \cos^2 \theta = \frac{1}{3} \] 
Step 3: Taking the square root of both sides: \[ \cos \theta = \frac{1}{\sqrt{3}} \] 
Step 4: Now, we know that: \[ \tan \theta = \frac{1}{\cos \theta} = \sqrt{2} \] Thus, the value of \( \tan \theta \) is \( \sqrt{2} \).