Question

Assertion :Eccentricity of rectangular hyperbola $xy=c^2$ is $\sqrt{2}$. Reason: The transverse & conjugate axes of rectangular hyperbola are equal.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is given by
$b^2=a^2(e^2-1)$ $\Rightarrow e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+{\left(\frac{2b}{2a}\right)}^2}=\sqrt{1+{\left(\frac{Conjugateaxis}{transverseaxis}\right)}^2}$ As the hyperbola is rectangular $\therefore$ Conjugate axis = transverse axis $\therefore e=\sqrt{1+1}=\sqrt{2}$
Hence, option 'A' is correct.