Question

Assertion :A is the vertex of a branch of the hyperbola. $x^2-2y^2-2\sqrt{2}x-4\sqrt{2}y-6=0$, B is one of the end points of its latus rectum and C is the focus of the hyperbola nearer to the point A.
Statement - 1: The area of the triangle ABC is $\sqrt{3/2}-1$ Reason: Statement - 2: Eccentricity of the hyperbola is $\sqrt{3/2}$ and length of the conjugate axis is 4.

• 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

Given hyperbola may be written as $\frac{(x-\sqrt{2}{)}^2}{4}-\frac{(y+\sqrt{2}{)}^2}{2}=1$
$\Rightarrow a=2,b=\sqrt{2}$
& eccentricity, $e=\sqrt{1+1/2}=\sqrt{\frac{3}{2}}$
Required area
$=\frac{1}{2}(ae-a)\times \frac{b^2}{a}=\frac{1}{2}\frac{(\sqrt{3}-\sqrt{2})\times 2}{\sqrt{2}}=\frac{(\sqrt{3}-\sqrt{2})}{\sqrt{2}}$
$\Rightarrow$ Area $=(\sqrt{\frac{3}{2}}-1)$ .
Hence, option 'A' is correct.