Question

The d.rs of two lines are $k,3,5$ and $2,-1,2$. If the angle between the lines is ${45}^o$, then $k=$

• A. $2$
• B. $3$
• C. $1$
• D. $4$

Answer 1

The correct option is D $4$

The dr's of the $2$ lines are given by,
$k,3,5$ and $2,-1,2$
The angle between them can be obtained by the dot product between them.
$\cos \theta =\frac{2k+7}{\left(3\right)\left(\sqrt{k^2+34}\right)}$
Since, $\theta ={45}^o$
$\frac{1}{\sqrt{2}}=\frac{2k+7}{\left(3\right)\left(\sqrt{k^2+34}\right)}$
Squaring on both sides
$\frac{1}{2}=\frac{4k^2+28K+49}{\left(9\right)\left(k^2+34\right)}$
$k^2-56K+208=0$
$(k-4)(k-52)=0$
$\therefore k=4$