Question

If the angle between the lines whose direction ratios are $(2,-1,2)$ and $(x,3,5)$ is $\frac{\pi }{4}$, then the smaller value of $x$ is

• A. $52$
• B. $4$
• C. $2$
• D. $1$

Answer 1

The correct option is B $4$

$\cos \frac{\pi }{4}=\frac{2x-3+10}{\sqrt{2^2+(-1{)}^2+2^2}\times \sqrt{x^2+3^2+5^2}}=\frac{2x+7}{3\times \sqrt{x^2+34}}$

\$⟹\frac{1}{\sqrt{2}}=\frac{2x+7}{3\times \sqrt{x^2+34}}$

$⟹\frac{1}{2}=\frac{(2x+7{)}^2}{9\times (x^2+34)}$

$⟹9x^2+306=8x^2+56x+98$

$⟹x^2-56x+208=0$

$⟹x=4,52$

The smaller value is 4.