Question
If $$x\,\cos \,\theta + y\,\sin \,\theta = 2$$ is perpendicular to the line $$x - y = 3,$$ then what is one of the value of $$\theta \,?$$
A.
$$\frac{\pi }{6}$$
B.
$$\frac{\pi }{4}$$
C.
$$\frac{\pi }{2}$$
D.
$$\frac{\pi }{3}$$
Answer :
$$\frac{\pi }{4}$$
Solution :
Consider a line
$$\eqalign{
& x\,\cos \,\theta + y\,\sin \,\theta = 2 \cr
& \Rightarrow y\,\sin \,\theta = - x\,\cos \,\theta + 2 \cr
& \Rightarrow y = - x\frac{{\cos \,\theta }}{{\sin \,\theta }} + \frac{2}{{\sin \,\theta }} \cr
& \Rightarrow y = - x\,\cot \,\theta + 2\,{\text{cosec}}\,\theta \cr} $$
On comparing this equation with $$y = mx + c$$ we get
slope of line $$x\,\cos \,\theta + y\,\sin \,\theta = 2{\text{ is }} - \cot \,\theta $$
Also, we have a line $$x - y = 3\, \Rightarrow y = x - 3$$
slope of line $$x - y = 3$$ is $$1$$
Since, both the lines are perpendicular to each other.
$$\therefore $$ Product of their slopes $$ = - 1$$
$$\eqalign{
& \Rightarrow \left( { - \cot \,\theta } \right)\left( 1 \right) = - 1 \cr
& \Rightarrow \cot \,\theta = 1 = \cot \frac{\pi }{4} \cr
& \Rightarrow \theta = \frac{\pi }{4} \cr} $$