Statement-1: If a hyperbola does not have mutually perpendicular tangents, then its eccentricity is greater than $\sqrt{2}$ .
Statement-2: Locus of a point from which two perpendicular tangents can be drawn to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ is
$x^{2} + y^{2} = a^{2} - b^{2}$

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STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is False, STATEMENT-2 is True

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is NOT a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

answer is A.

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Detailed Solution

tatement-2 is true as $x^{2} + y^{2} = a^{2} - b^{2}$ is the

director circle of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ using it in

statement-1, if the hyperbola does not have mutually perpendicular tangents then the points on $x^{2} + y^{2} = a^{2} - b^{2}$ are

not real.

$\Rightarrow b^{2} > a^{2} \Rightarrow e^{2} = \frac{a^{2} - b^{2}}{a^{2}} > 2 \Rightarrow e > \sqrt{2}$ and thus statement-1 is also true.

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