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  • If the chord joining two points <img alt="\mathrm{\left(a \sec \theta_1, b \tan \theta_1\right)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cleft%28a%20%5Csec%20%5Ctheta_1%

If the chord joining two points \mathrm{\left(a \sec \theta_1, b \tan \theta_1\right)} and \mathrm{\left(a \sec \theta_2, b \tan \theta_2\right)} passes through the focus of the hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}, then \mathrm{\tan \frac{\theta_1}{2} \tan \frac{\theta_2}{2}=}

Option: 1

\frac{1-e}{1+e}


Option: 2

\frac{e-1}{e+1}


Option: 3

\frac{e+1}{e-1}


Option: 4

\frac{1+e}{1-e}


Answers (1)

best_answer

The equation of the chord joining \left(a \sec \theta_1, b \tan \theta_1\right) and \left(a \sec \theta_2, b \tan \theta_2\right) is

\frac{x}{a} \cos \left(\frac{\theta_1-\theta_2}{2}\right)-\frac{y}{b} \sin \left(\frac{\theta_1+\theta_2}{2}\right)=\cos \left(\frac{\theta_1+\theta_2}{2}\right)

If it passes through the focus (a e, 0), then

\begin{aligned} & \qquad \cos \left(\frac{\theta_1-\theta_2}{2}\right)=\cos \left(\frac{\theta_1+\theta_2}{2}\right) \\\\ & \Rightarrow \frac{\cos \left(\frac{\theta_1-\theta_2}{2}\right)}{\cos \left(\frac{\theta_1+\theta_2}{2}\right)}=\frac{1}{e} \\\\ & \Rightarrow \quad \frac{\cos \left(\frac{\theta_1-\theta_2}{2}\right)+\cos \left(\frac{\theta_1+\theta_2}{2}\right)}{2}=\frac{1+e}{1-e} \\\\ & \Rightarrow \cos \left(\frac{\theta_1-\theta_2}{2}\right)-\cos \left(\frac{\theta_1+\theta_2}{2}\right) \\\\ & \cot \frac{\theta_1}{2} \cot \frac{\theta_2}{2}=\frac{1+e}{1-e} \Rightarrow \tan \frac{\theta_1}{2} \tan \frac{\theta_2}{2}=\frac{1-e}{1+e} \end{aligned}

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