First slide

Hyperbola in conic sections

Question

Statement-1: The locus of the point of intersection of the tangents that are at right angles to the hyperbola $\frac{x^{2}}{36} - \frac{y^{2}}{16} = 1 is the circle x^{2} + y^{2} = 52$
Statement-2: Perpendicular tangents to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ interest on the director circle $x^{2} + y^{2} = a^{2} - b^{2} (a^{2} > b^{2})$ of the hyperbola.

Moderate

A

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

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

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

Equation of a tangent to the hyperbola is $y = m x$ + $\sqrt{a^{2} m^{2} - b^{2}}$ and the equation of the tangent perpendicular to it is $y = - \frac{1}{m} x + \sqrt{\frac{a^{2}}{m^{2}} - b^{2}}$ .
Eliminating m we get the required locus as $(y - m x)^{2}$ $+ (x + m y)^{2} = a^{2} m^{2} - b^{2} + a^{2} - b^{2} m^{2}$
$\Rightarrow x^{2} + y^{2} = a^{2} - b^{2}$ and the statement-2 is true using which statement-1 is false.

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