Question

Let α be the angle between the lines whose direction cosines satisfy the equations l + m - n = 0 and l² + m² - n² = 0. Then the value of sin⁴α + cos⁴α is :

• A. 5/8
• B. 3/4
• C. 1/2
• D. 3/4

Hint:

Always simplify the linear equation first and substitute it into the quadratic one to find the specific ratios for both lines.

Answer 1

The Correct Option is A

Step 1: From $n = l + m$, substitute into $l^2 + m^2 - (l+m)^2 = 0 \Rightarrow -2lm = 0$.
Step 2: Case 1: $l=0 \Rightarrow n=m$. DRs are $(0, 1, 1)$. Case 2: $m=0 \Rightarrow n=l$. DRs are $(1, 0, 1)$.
Step 3: $\cos \alpha = \frac{|0+0+1|}{\sqrt{2}\sqrt{2}} = 1/2 \Rightarrow \alpha = 60^\circ$.
Step 4: $\sin^4 60^\circ + \cos^4 60^\circ = (\frac{3}{4})^2 + (\frac{1}{4})^2 = \frac{9}{16} + \frac{1}{16} = \frac{10}{16} = 5/8$.