Search any question & find its solution
Question:
Answered & Verified by Expert
If the pair of lines given by $\left(x^2+y^2\right) \cos ^2 \theta=(x \cos \theta+y \sin \theta)^2 \quad$ are perpendicular to each other, then $\theta$ is equal to
Options:
Solution:
2158 Upvotes
Verified Answer
The correct answer is:
$\frac{\pi}{4}$
Given pair of lines is
$\begin{aligned} & \left(x^2+y^2\right) \cos ^2 \theta=(x \cos \theta+y \sin \theta)^2 \\ \Rightarrow & \left(x^2+y^2\right) \cos ^2 \theta=x^2 \cos ^2 \theta+y^2 \sin ^2 \theta \\ & +2 x y \sin \theta \cos \theta \\ \Rightarrow & y^2\left(\cos ^2 \theta-\sin ^2 \theta\right)-2 x y \sin \theta \cos \theta=0\end{aligned}$
$\because$ Pair of lines are perpendicular to each other.
$\therefore$ Coefficient of $x^2+$ Coefficient of $y^2=0$
$\begin{array}{ll}\therefore & 0+\cos ^2 \theta-\sin ^2 \theta=0 \\ \Rightarrow & \tan ^2 \theta=1 \\ \Rightarrow & \tan \theta= \pm 1 \Rightarrow \quad \theta=\frac{\pi}{4} \text { or } \frac{3 \pi}{4}\end{array}$
$\begin{aligned} & \left(x^2+y^2\right) \cos ^2 \theta=(x \cos \theta+y \sin \theta)^2 \\ \Rightarrow & \left(x^2+y^2\right) \cos ^2 \theta=x^2 \cos ^2 \theta+y^2 \sin ^2 \theta \\ & +2 x y \sin \theta \cos \theta \\ \Rightarrow & y^2\left(\cos ^2 \theta-\sin ^2 \theta\right)-2 x y \sin \theta \cos \theta=0\end{aligned}$
$\because$ Pair of lines are perpendicular to each other.
$\therefore$ Coefficient of $x^2+$ Coefficient of $y^2=0$
$\begin{array}{ll}\therefore & 0+\cos ^2 \theta-\sin ^2 \theta=0 \\ \Rightarrow & \tan ^2 \theta=1 \\ \Rightarrow & \tan \theta= \pm 1 \Rightarrow \quad \theta=\frac{\pi}{4} \text { or } \frac{3 \pi}{4}\end{array}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.