Question

Acute angle bwtween the lines $ax+by+c=0$ and $x\cos \theta +y\sin \theta =c(c\ne 0)$ is ${45}^{\circ }.$ If both the lines meet with the line $y\cos \theta =x\sin \theta$ at same point, then the value of $a^2+b^2$ is

• A. $2$
• B. $1$
• C. $4$
• D. $3$

Answer 1

The correct option is A $2$

All the three lines are concurrent
$\therefore \left|\begin{array}{ccc}a& b& c\\ \cos \theta & \sin \theta & -c\\ \sin \theta & -\cos \theta & 0\end{array}\right|=0\Rightarrow -ac\cos \theta -bc\sin \theta -c=0\Rightarrow a\cos \theta +b\sin \theta =-1\cdots \left(1\right)$

$\tan {45}^{\circ }=\left|\frac{-\frac{a}{b}+\cot \theta }{1+\frac{a}{b}\cot \theta }\right|\Rightarrow |1+\frac{a}{b}\cot \theta |=|-\frac{a}{b}+\cot \theta |\Rightarrow |b\sin \theta +a\cos \theta |=|-a\sin \theta +b\cos \theta |\Rightarrow |-a\sin \theta +b\cos \theta |=1\cdots \left(2\right)(\because \text{eq.}(1\left)\right)$

Squaring and adding $\left(1\right)$ and $\left(2\right),$ we get
$a^2+b^2=2$