The point of the contact of $9 x + 8 y - 11 = 0$ to the hyperbola $3 x^{2} - 4 y^{2} = 11$ is

(3, - 2)

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(3, 2)

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(- 3, - 3)

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(3, 3)

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Solution

The correct option is B $(3, - 2)$
Equation of hyperbola

$3 x^{2} - 4 y^{2} = 11$ - (1)

Equation of line,

$9 x + 8 y - 11 = 0$
$y = \frac{11 - 9 x}{8}$ - (2)

Put (2) in (1)
$3 x^{2} - 4 \frac{{(11 - 9 x^{2})}^{2}}{(8)^{2}} = 11$

$3 x^{2} - \frac{4}{64} (121 + 81 x^{2} - 198 x) = 11$
$3 x^{2} - \frac{121}{16} - \frac{81}{16} x^{2} + \frac{198}{16} x = 11$

$x^{2} (\frac{48 - 81}{16}) + \frac{99}{8} x - \frac{297}{16} = 0$

$- 33 x^{2} + 198 x - 297 = 0$

$33 x^{2} - 198 x + 297 = 0$
$3 x^{2} - 18 x + 27 = 0$

$x^{2} - 6 x + 9 = 0$
$x = 3, 3$
So, $y = - 2$
Point $= (3, - 2)$

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