Question

The angle between the lines represented by the equation $\left(x^2+y^2\right)\sin \left(\theta \right)+2xy=0$ is

• A. $\theta$
• B. $\frac{\theta }{2}$
• C. $\frac{\mathrm{\pi }}{2}-\theta$
• D. $\frac{\mathrm{\pi }}{2}-\frac{\theta }{2}$

Answer 1

The correct option is C

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

Explanation for correct answer:

Finding the angle between the lines:

Given the equation $\left(x^2+y^2\right)\sin \left(\theta \right)+2xy=0$

$\Rightarrow x^2\sin \left(\theta \right)+2xy+y^2\sin \left(\theta \right)=0$

Now comparing the given equation with the general form of an equation of pair of lines $a{x}^2+2hxy+b{y}^2=0$

We get $a=\sin \left(\theta \right)$,$b=\sin \left(\theta \right)$,$2h=2\Rightarrow h=1$

Finding the angle between the two lines using the formula

$\tan \beta =\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|$

Substituting $a=\sin \left(\theta \right)$ ,$b=\sin \left(\theta \right)$ and $h=1$in the above equation. we get

$$\begin{gathered} \tan \left(\beta \right)=\left|\frac{2\sqrt{1-\left(\sin \left(\theta \right)\right)\left(\sin \left(\theta \right)\right)}}{\sin \left(\theta \right)+\sin \left(\theta \right)}\right| \\ =\left|\frac{2\sqrt{1-{\left(\sin \left(\theta \right)\right)}^2}}{2\sin \left(\theta \right)}\right| \\ =\left|\frac{\sqrt{1-{\sin }^2\left(\theta \right)}}{\sin \left(\theta \right)}\right| \end{gathered}$$

Substituting $1-{\sin }^2\left(\theta \right)={\cos }^2\left(\theta \right)$

$$\begin{gathered} \tan \left(\beta \right)=\left|\frac{\sqrt{{\cos }^2\left(\theta \right)}}{\sin \left(\theta \right)}\right| \\ =\left|\frac{c\mathrm{os}\left(\theta \right)}{\sin \left(\theta \right)}\right| \\ =\left|\cot \left(\theta \right)\right| \end{gathered}$$

Removing the modulus

$$\begin{gathered} tan\left(\mathrm{\beta }\right)=\text{cot}\left(\theta \right) \\ =\text{tan}(90-\theta ) \end{gathered}$$

Apply ${\tan }^{-1}$ on both sides

$$\begin{gathered} {\tan }^{-1}\left(\tan \left(\mathrm{\beta }\right)\right)={\text{tan}}^{-1}\left(\text{tan}(90-\theta )\right) \\ \mathrm{\beta }=90-\mathrm{\theta } \end{gathered}$$

Therefore, the angle between the two lines is $90-\mathrm{\theta }$

Hence, option (C) is the correct answer