Question

The straight lines whose DC's are $l_i,m_i,n_i$ which are roots of $al+bm+cn=0$ and $f{l}^2+g{m}^2+h{n}^2=0$ are parallel

• A. $\frac{f}{a}+\frac{g}{b}+\frac{h}{c}=0$
• B. $\frac{a}{f}+\frac{b}{g}+\frac{c}{h}=0$
• C. $\frac{f}{a^2}+\frac{g}{b^2}+\frac{h}{c^2}=0$
• D. $\frac{a^2}{f}+\frac{b^2}{g}+\frac{c^2}{h}=0$

Answer 1

Answer: (D)

$\frac{a^2}{f}+\frac{b^2}{g}+\frac{c^2}{h}=0$

Direction cosines of the two lines are given by $al+bm+cn=0$ ------$\left(1\right)$
and $f{l}^2+g{m}^2+h{n}^2=0$ ------$\left(2\right)$
Eliminating $n$ from $\left(1\right)$ and $\left(2\right)$ gives
$f{l}^2+g{m}^2+h{\left(\frac{al+bm}{-c}\right)}^2=0$
$\Rightarrow c^{2}f{l}^2+c^{2}g{m}^2+c(al+bm{)}^2=0$
$\Rightarrow (f{c}^2+h{a}^2){\left(\frac{l}{m}\right)}^2+2abh\left(\frac{l}{m}\right)+(g{c}^2+h{b}^2)=0$
$\frac{l_1}{m_1}$ 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 $0$.
$\Rightarrow 4a^2{b}^2{h}^2=4(f{c}^2+h{a}^2)(g{c}^2+h{b}^2)$
$\Rightarrow fg{c}^4+fh{c}^2{b}^2+gh{a}^2{c}^2=0$
$\Rightarrow fg{c}^2+hf{b}^2+gh{a}^2=0$
$\therefore \frac{a^2}{f}+\frac{b^2}{g}+\frac{c^2}{h}=0$
Hence, option D.