The point of intersection of two tangents dra | vedclass

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(D) The locus of the point of intersection of two perpendicular tangents to a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is its director circle

Vedclass · Accepted Answer

$3$

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(D) The locus of the point of intersection of two perpendicular tangents to a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is its director circle,given by the equation $x^2 + y^2 = a^2 - b^2$.Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{4} = 1$,we have $b^2 = 4$.The director circle is $x^2 + y^2 = a^2 - 4$.Comparing this with the given circle $x^2 + y^2 = 5$,we get $a^2 - 4 = 5$.$a^2 = 9 \Rightarrow a = 3$ (since $a > 0$ for a hyperbola).

The point of intersection of two tangents drawn to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{4} = 1$ lies on the circle $x^2 + y^2 = 5$. If these tangents are perpendicular to each other,then $a =$

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