Question

Suppose direction cosines of two lines are given by $ul+vm+wn=0$ and $a{l}^2+b{m}^2+c{n}^2=0$ where $u,v,w,a,b,c$ are arbitrary constants and $l,m,n$ are direction cosines of the line. The given lines will be parallel if

• A. $\sum u^2(b+c)=0$
• B. $\sum \frac{a^2}{u}=0$
• C. $\sum \frac{u^2}{a}=0$
• D. $\sum \frac{(b+c)}{u^2}=0$

Answer 1

The correct option is C $\sum \frac{u^2}{a}=0$

Direction cosines of the two lines are given by $ul+vm+wn=0$ ------(1)
and $a{l}^2+b{m}^2+c{n}^2=0$ ------(2)
Eliminating $n$ from (1) and (2) gives
$a{l}^2+b{m}^2+c{\left(\frac{ul+vm}{-w}\right)}^2=0$
$\Rightarrow w^{2}a{l}^2+w^{2}b{m}^2+c(ul+vm{)}^2=0$
$\Rightarrow (a{w}^2+c{u}^2){\left(\frac{l}{m}\right)}^2+2uvc\left(\frac{l}{m}\right)+(b{w}^2+c{v}^2)=0$
$\frac{l_1}{m_1}$ and $\frac{l_2}{m_2}$ are roots of above equation, if lines are parallel then direction cosines are equal.
i.e discriminant value of above quadratic equation is $0$.
$\Rightarrow 4u^2{v}^2{c}^2=(a{w}^2+c{u}^2)(b{w}^2+c{v}^2)$
$\Rightarrow ab{w}^4+ac{w}^2{v}^2+bc{u}^2{w}^2=0$
$\Rightarrow ab{w}^2+ac{v}^2+bc{u}^2=0$
$\therefore \frac{u^2}{a}+\frac{v^2}{b}+\frac{w^2}{c}=0$
Hence, option C.