Question

Assertion :A hyperbola whose asymptotes are at right angles to each other is called rectangular hyperbola. Reason: Eccentricity of rectangular hyperbola is $\sqrt{2}$.

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

The equation of the asymptotes to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are given by
$y=\pm \frac{b}{a}xi.e.y=\frac{b}{a}x$ & $y=\frac{-b}{a}x$ If $\theta$ be the angle between the asymptotes, then
$\tan \theta =\frac{\frac{b}{a}-(-\frac{b}{a})}{1+\left(\frac{b}{a}\right)(-\frac{b}{a})}$ $\Rightarrow \tan \theta =\frac{2ab}{a^2-b^2}$ Putting $\theta ={90}^{\circ }\Rightarrow a=b$ $\Rightarrow$ hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ converted to $x^2-y^2=a^2$ and the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is given by $e^2=1+\frac{b^2}{a^2}=1+\frac{a^2}{a^2}=2\Rightarrow e=\sqrt{2}$
As Assertion (A) & Reason (R) both are true but Reason (R) is not the correct explanation of Assertion (A).
Hence, option 'B' is correct.