Question

If $\theta$ is the angle between the lines $a{x}^2+2bxy+b{y}^2=0$, then angle between $x^2+2xysec\theta +y^2=0$ is:

• A. $\theta$
• B. $2\theta$
• C. $\frac{\theta }{2}$
• D. $3\theta$

Answer 1

The correct option is A

$\theta$

Explanation for the correct option.

The general equation of two lines is $a{x}^2+2hxy+b{y}^2=0$ and the angle between these two lines is given by $\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathbf{}\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|$.

Comparing the given equation $x^2+2xysec\theta +y^2=0$ with the general equation, we get $a=1,h=sec\theta ,b=1$, thus the angle between the lines will be:

$$\begin{gathered} ={\tan }^{-1}\left(\frac{2\sqrt{se{c}^2\theta -1}}{2}\right) \\ ={\tan }^{-1}\left(\sqrt{{\tan }^2\theta }\right) \\ ={\tan }^{-1}\left(\tan \theta \right) \\ =\theta \end{gathered}$$

Hence, option A is correct.