Question

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

The equation of the given pair of straight lines is $x^2+2xysec\theta +y^2=0$

Comparing this equation with the general equation of a pair of straight lines $a{x}^2+2hxy+b{y}^2=0$ we get

$a=1$, $h=sec\theta$, $b=1$

It is also known that the angle between the pair of straight lines $a{x}^2+2hxy+b{y}^2=0$ is given as

$\theta ={\tan }^{-1}\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|$

Let $\alpha$ be the angle between the lines $x^2+2xysec\theta +y^2=0$

Substituting the values of $a,h$ and $b$ we get

$\Rightarrow$$\alpha ={\tan }^{-1}\left|\frac{2\sqrt{se{c}^2\theta -1\times 1}}{1+1}\right|$

$\Rightarrow$ $={\tan }^{-1}\left|\sqrt{se{c}^2\theta -1}\right|$

$\Rightarrow$ $={\tan }^{-1}\left|\sqrt{{\tan }^2\theta }\right|$

$\Rightarrow$ $={\tan }^{-1}\left(\tan \theta \right)$

$\Rightarrow$ $\alpha =\theta$

Hence, the angle between the lines $x^2+2xysec\theta +y^2=0$ is $\theta$.

So, option $\left(A\right)$ is the correct answer.