Question

$PQ$ is the chord joining the points ${\theta }_1$ and ${\theta }_2$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If ${\theta }_1-{\theta }_2=2\alpha ,\alpha \ne \frac{(2n+1)\pi }{2}$ . If $PQ$ touches the hyperbola $\frac{x^2}{A^2}-\frac{y^2}{b^2}=1$ then :

• A. Length of transverse axis $=2a\sec \alpha \text{units}$
• B. Length of transverse axis $=2a\cos \alpha \text{units}$
  
• C. $e=\sqrt{1+\frac{b^2{\cos }^2\alpha }{a^2}}$
• D. $e=\sqrt{1+\frac{b^2{\sec }^2\alpha }{a^2}}$

Answer 1

Answer: (A), (C)

The correct options are
A Length of transverse axis $=2a\sec \alpha \text{units}$
C $e=\sqrt{1+\frac{b^2{\cos }^2\alpha }{a^2}}$

The equation of chord $PQ$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

$\frac{x}{a}\cos \left(\frac{{\theta }_1-{\theta }_2}{2}\right)-\frac{y}{b}\sin \left(\frac{{\theta }_1+{\theta }_2}{2}\right)=\cos \left(\frac{{\theta }_1+{\theta }_2}{2}\right)\Rightarrow \frac{x}{a}\cos \alpha -\frac{y}{b}\sin \left(\frac{{\theta }_1+{\theta }_2}{2}\right)=\cos \left(\frac{{\theta }_1+{\theta }_2}{2}\right)(\because {\theta }_1-{\theta }_2=2\alpha )\Rightarrow y=\frac{xb\cos \alpha }{a\sin \left(\frac{{\theta }_1+{\theta }_2}{2}\right)}-b\cot \left(\frac{{\theta }_1+{\theta }_2}{2}\right)\cdots \left(i\right)$

Now if equation $\left(i\right)$ is tangent to $\frac{x^2}{A^2}-\frac{y^2}{b^2}=1,$ then we have

$c^2=A^2\cdot m^2-b^2\Rightarrow b^2{\cot }^2\left(\frac{{\theta }_1+{\theta }_2}{2}\right)=A^2{m}^2-b^2\Rightarrow b^2{\text{cosec}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}\right)=A^2{m}^2\therefore A^2=\frac{a^2}{{\cos }^2\alpha }$

Hence, the length of transverse axis
$=2A=2a\sec \alpha \text{units}$
$\Rightarrow e=\sqrt{1+\frac{b^2{\cos }^2\alpha }{a^2}}$