Locus of point of intersection of any two perpendicular tangents to hyperbola is a circle which is c

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10-2. Parabola, Ellipse, Hyperbola

easy

The locus of the point of intersection of any two perpendicular tangents to the hyperbola is a circle which is called the director circle of the hyperbola, then the eqn of this circle is

A

${x^2} + {y^2} = {a^2} + {b^2}$

B

${x^2} + {y^2} = {a^2} - {b^2}$

C

${x^2} + {y^2} = 2ab$

D

None of these

Solution

(b) Equation of hyperbola is $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$

Any tangent to hyperbola are $y = mx \pm \sqrt {{a^2}{m^2} – {b^2}} $

Also tangent perpendicular to this is $y = \frac{{ – 1}}{m}x \pm \sqrt {\frac{{{a^2}}}{{{m^2}}} – {b^2}} $

Eliminating $m$, we get ${x^2} + {y^2} = {a^2} – {b^2}$.

Standard 11

Mathematics

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