Question

If the direction cosines of a line are \[ \left( \frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}} \right) \] and \(c-a=4\), then \(ca=\)

• A. \(24\)
• B. \(21\)
• C. \(18\)
• D. \(33\)

Hint:

For direction cosines \((l,m,n)\), always remember: \[ l^2+m^2+n^2=1 \] This identity is very useful for finding unknown quantities in 3D geometry problems.

Answer 1

The Correct Option is B

Step 1: Apply the direction cosine identity.
$l^2+m^2+n^2=1$ with $l=a/\sqrt{83}$, $m=5/\sqrt{83}$, $n=c/\sqrt{83}$ gives \[a^2+25+c^2=83 \implies a^2+c^2=58.\]
Step 2: Use $c-a=4$.
$c=a+4$. Substituting: $a^2+(a+4)^2=58 \implies 2a^2+8a-42=0$.
Step 3: Solve the quadratic.
$a^2+4a-21=0 \implies (a+7)(a-3)=0$. So $a=3$ or $a=-7$.
Step 4: Find $c$ and product $ca$.
$a=3 \implies c=7$, $ca=21$. $a=-7 \implies c=-3$, $ca=21$. Both cases give $ca=21$.
Step 5: Verify.
$a=3,c=7$: $a^2+c^2=58$ and $c-a=4$. Both checks pass.
Step 6: State the answer.
\[ \boxed{ca=21} \]