Question

The angle between the pair of straight lines $y^2{\sin }^2\theta -xy{\sin }^2\theta +x^2({\cos }^2\theta -1)=0$ is

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{\mathrm{\pi }}{6}$
• D. $\frac{\mathrm{\pi }}{2}$

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct option:

Given equation:

$y^2{\sin }^2\theta -xy{\sin }^2\theta +x^2({\cos }^2\theta -1)=0$

On comparing this with the general equation $a{x}^2+b{y}^2+2hxy=0$, we get

$$\begin{gathered} a={\cos }^2\theta -1 \\ b={\sin }^2\theta \end{gathered}$$

We know that when pair of lines are perpendicular then the coefficient of $x^2$and $y^2$ sums up to $1$.

$$\begin{gathered} a+b=0 \\ \Rightarrow {\cos }^2\theta -1+{\sin }^2\theta =0\mathbf{[}\mathbf{\because }\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{\theta }\mathbf{+}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{\theta }\mathbf{=}\mathbf{1}] \\ \Rightarrow 1-1=0 \end{gathered}$$

So, we see that condition for perpendicular lines is true for given values,

Therefore, both pairs of straight lines are perpendicular to each other.

Hence, the correct option is (D)