Question

If the direction cosines of a line are: $$ \left( \frac{a}{\sqrt{83}},\ \frac{5}{\sqrt{83}},\ \frac{c}{\sqrt{83}} \right) $$ and it is given that $ c - a = 4 $, find $ ca $.

• A. 24
• B. 21
• C. 18
• D. 33

Hint:

Use the identity \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \) for direction cosines, and substitute constraints.

Answer 1

The Correct Option is B

For direction cosines, their squares add to 1: \[ \left( \frac{a}{\sqrt{83}} \right)^2 + \left( \frac{5}{\sqrt{83}} \right)^2 + \left( \frac{c}{\sqrt{83}} \right)^2 = 1 \Rightarrow \frac{a^2 + 25 + c^2}{83} = 1 \Rightarrow a^2 + c^2 = 58 \quad \text{(1)} \] Also given: \[ c - a = 4 \Rightarrow c = a + 4 \quad \text{(2)} \] Substitute (2) into (1): \[ a^2 + (a + 4)^2 = 58 \Rightarrow a^2 + a^2 + 8a + 16 = 58 \Rightarrow 2a^2 + 8a - 42 = 0 \Rightarrow a^2 + 4a - 21 = 0 \] Solving: \[ a = \frac{-4 \pm \sqrt{16 + 84}}{2} = \frac{-4 \pm \sqrt{100}}{2} = \frac{-4 \pm 10}{2} \Rightarrow a = 3,\ -7 \Rightarrow c = 7,\ -3 \] Now \( ca = 3 \cdot 7 = \boxed{21} \)