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. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. None of these.

Answer 1

The correct option is A

$\frac{\pi }{2}$

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

Given: $y^2{\sin }^2\theta -xy{\sin }^2\theta +x^2({\cos }^2\theta -1)=0,$
$$\begin{gathered} y^2{\sin }^2\theta -xy{\sin }^2\theta +x^2({\cos }^2\theta -1)=0 \\ {y}^2{\sin }^2\theta -xy{\sin }^2\theta -x^2{\sin }^2\theta =0 \end{gathered}$$
we know that, for $a{x}^2+b{y}^2+2hxy=0\text{if}a+b=0$ then the pair of lines are perpendicular

Here,

$$\begin{gathered} a=-{\sin }^2\theta , \\ b={\sin }^2\theta \\ \Rightarrow a+b=0 \end{gathered}$$
Therefore, the lines are perpendicular, i.e. the angle between the given pair of straight line is $\frac{\pi }{2}$

Hence, the correct answer is option (A)