Study the following statement and select the correct option:
A is the vertex of a branch of the hyperbola $x 2 − 2 y 2 − 22 x − 42 y − 6 = 0, B$ is one of the end point of its latusrectum and C is the focus of the hyperbola nearer to the point A
Statement-I : The area of the triangle ABC is $32 − 1$
statement-II: Eccentricity of the hyperbola is $22$ and length of the conjugate axis is 4.

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Statement-1 is true, statement-2 is true , statement-2 is correct explanation of statement- 1

Statement-1 is true, statement-2 is true , statement-2 is not correct explanation of statement-1

Statement-1 is true, statement-2 is false

Statement-1 is false, statement-2 is false

answer is C.

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Detailed Solution

$x 2 − 2 y 2 − 22 x − 42 y − 6 = 0 (x − 2) 2 4 − (y + 2) 2 2 = 1$
$e = \sqrt{\frac{4 + 2}{4}} = \sqrt{\frac{3}{2}} A C = a e - a = a (e - 1), B C = \frac{b^{2}}{a} = semi latus rectum$
Area of $△ A B C = 12 a (e − 1) ⋅ b 2 a = 32 − 1$
Length of conjugate axis $= 2 b = 22$