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If $\varphi^{\prime}$ is the angle between the lines $a x^{2}+2 h x y+b y^{2}=0$, then angle between $x^{2}+2 x y \sec \theta+y^{2}=0$ is
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The correct answer is:
$\bar{\theta}$
Angle between the lines $a x^{2}+2 h x y+b y^{2}=0$ is $\tan \theta=\left|\frac{2 \sqrt{h^{2}-a b}}{a+b}\right|$
For $x^{2}+2 x y \sec \theta+y^{2}=0$
$\begin{aligned} h &=\sec \theta, a=b=1 \\ \therefore \quad \tan \phi &=\mid \frac{2 \sqrt{\sec ^{2} \theta-1}}{1+1} \end{aligned}$
$=\frac{2 \tan \theta}{2}=\tan \theta$
$\therefore$ Angle between $x^{2}+2 x y \sec \theta+y^{2}=0$ is $\theta$.
For $x^{2}+2 x y \sec \theta+y^{2}=0$
$\begin{aligned} h &=\sec \theta, a=b=1 \\ \therefore \quad \tan \phi &=\mid \frac{2 \sqrt{\sec ^{2} \theta-1}}{1+1} \end{aligned}$
$=\frac{2 \tan \theta}{2}=\tan \theta$
$\therefore$ Angle between $x^{2}+2 x y \sec \theta+y^{2}=0$ is $\theta$.
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