The number of pairs of perpendicular tangents to the hyperbola $3 x^{2} - 6 x - 2 y^{2} + 4 y - 23 = 0$ is:

0

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1

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2

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infinitely many

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Solution

The correct option is A $0$
Equation of the hyperbola is
$3 x^{2} - 6 x - 2 y^{2} + 4 y - 23 = 0$
$\Rightarrow 3 (x - 1)^{2} - 2 (y - 1)^{2} = 24$
$\Rightarrow \frac{(x - 1)^{2}}{8} - \frac{(y - 1)^{2}}{12} = 1$
Equation of director circle is
$(x - 1)^{2} + (y - 1)^{2} = 8 - 12 = - 4$ which is not possible
Thus number of pairs of perpendicular tangents is $0$

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