Question

The angle between the straight lines, whose direction cosines are given by the equations $2l+2m−n=0$ and $mn+nl+lm=0$, is :

• A. $\frac{Ï€}{2}$
• B. $π−{\cos }^{−1}\left(\frac{4}{9}\right)$
• C. ${\cos }^{−1}\left(\frac{8}{9}\right)$
• D. $\frac{Ï€}{3}$

Answer 1

Answer: (A)

$n=2(â„“+m)$
$â„“m+n(â„“+m)=0$
$â„“m+2(â„“+m{)}^2=0$
$2{â„“}^2+2m^2+5mâ„“=0$
$2{\left(\frac{â„“}{m}\right)}^2+2+5\left(\frac{â„“}{m}\right)=0$
$2t^2+5t+2=0$
$(t+2)(2t+1)=0$
$⇒t=−2;−\frac{1}{2}$
(i) $\frac{ℓ}{m}=−2$
$\frac{n}{m}=−2$
$(−2m,m,−2m)$
$(−2,1,−2)$
(ii) $\frac{ℓ}{m}=−\frac{1}{2}$
$n=−2ℓ$
$(ℓ,−2ℓ,−2ℓ)$
$(1,−2,−2)$
$\cos θ=\frac{−2−2+4}{\sqrt{9}\sqrt{9}}=0$
$⇒0=\frac{π}{2}$