Question

If $\theta$ is the angle between the lines whose DR's are $(1,-2,1),(4,3,2),$ then $\sec \frac{\theta }{2}+\text{cosec}\frac{\theta }{2}=$

• A. $2\sqrt{2}$
• B. $\infty$
• C. $\sqrt{2}$
• D. $\frac{1}{2\sqrt{2}}$

Answer 1

The correct option is A $2\sqrt{2}$

Given: DR's $(1,-2,1),(4,3,2)$
i.e. $(1,-2,1)=(a_1,b_1,c_1)$ and
$(4,3,2)=(a_2,b_2,c_2)$
For angle $\theta$ Finding value of
$\Rightarrow a_1{a}_2+b_1{b}_2+c_1{c}_2$
$\Rightarrow 1\times 4+(-2)\times 3+1\times 2$
$\Rightarrow 4-6+2$
$\Rightarrow 0$
Since, $a_1{a}_2+b_1{b}_2+c_1{c}_2=0$ Hence, $\theta =\frac{\pi }{2}$
Now finding, $\sec \frac{\theta }{2}+\text{cosec}\frac{\theta }{2}=\sec \frac{\pi }{4}+\text{cosec}\frac{\pi }{4}=\sqrt{2}+\sqrt{2}$
$\Rightarrow \sec \frac{\theta }{2}+\text{cosec}\frac{\theta }{2}=2\sqrt{2}$