Question

Assertion :Eccentricity of the hyperbola $3x^2-6y^2=-18$ is $\sqrt{3}$. Reason: Eccentricity of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is derived by the relation $b^2=a^2(e^2-1)$.

• 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

$e=\sqrt{1+{\left(\frac{b}{a}\right)}^2}$
$\therefore e^2=1+\frac{b^2}{a^2}\Rightarrow b^2=a^2(e^2-1)$
Reason (R) is true
Now $3x^2-6y^2=-18\Rightarrow \frac{x^2}{6}-\frac{y^2}{3}=-1$
which is of form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$ where $a^2=6,b^2=3$
$\therefore$ Length of transverse axis $=2b=2\sqrt{3}$ and Length of conjugate axis $=2a=2\sqrt{6}$
$\therefore e=\sqrt{1+\frac{a^2}{b^2}}=\sqrt{1+\frac{6}{3}}=\sqrt{3}$
Thus both Assertion and Reason are true but Assertion is not following Reason.
Hence, option 'B' is correct.