Question

If the direction cosines of a line $L$ are $(ab,b,b)$ and the angle between $L$ and $X$ axis is $\frac{\pi }{3}$. Then the value of $\frac{1}{a^2}+\frac{1}{b^2}$ is

• A. $\frac{25}{6}$
• B. $\frac{6}{25}$
• C. $\frac{25}{8}$
• D. $\frac{35}{6}$

Answer 1

The correct option is A $\frac{25}{6}$

Given: DC's of Line $L$ $(l,m,n)=(ab,b,b).$
DC's of $x$ axis: $(1,0,0)$
Angle between $L$ and $x$ axis is $\frac{\pi }{3}$
i.e. $\cos \theta =l_1{l}_2+m_1{m}_2+n_1{n}_2$
$\Rightarrow \cos \left(\frac{\pi }{3}\right)=ab+0+0$
$\Rightarrow \frac{1}{2}=ab$
Also we know,
$l^2+m^2+n^2=1$
$(ab{)}^2+b^2+b^2=1\Rightarrow 2b^2=1-\frac{1}{4}=\frac{3}{4}$
$\Rightarrow b^2=\frac{3}{8}\Rightarrow b=\pm \sqrt{\frac{3}{8}}$
We have:
$ab=\frac{1}{2}\Rightarrow a=\pm \sqrt{\frac{2}{3}}$
Therefore, $\frac{1}{a^2}+\frac{1}{b^2}=\frac{3}{2}+\frac{8}{3}=\frac{25}{6}$