First slide

Hyperbola in conic sections

Question

$If x = 9 is the chord of contact of the hyperbola x^{2} - y^{2} = 9, then the equation of the corresponding pair of$ $tangents is$

Moderate

A

$9 x^{2} - 8 y^{2} + 18 x - 9 = 0$
B

$9 x^{2} - 8 y^{2} - 18 x + 9 = 0$
C

$9 x^{2} - 8 y^{2} - 18 x - 9 = 0$
D

$9 x^{2} - 8 y^{2} + 18 x + 9 = 0$

Solution

$Let a pair of tangents be drawn from the point (x_{1}, y_{1}) to the hyperbola$
$x^{2} - y^{2} = 9$
Then the chord of contact will be
${xx}_{1} - {yy}_{1} = 9 - - - - - (i)$
But the given chord of contact is
$x = 9 - - - - - (i i)$
As (i) and (ii) represent the same line, these equations should be identical and, hence
$\frac{x_{1}}{1} = - \frac{y_{1}}{0} = \frac{9}{9} or x_{1} = 1, y_{1} = 0$
Therefore, the equation of pair of tangents drawn from (1,0) to $x^{2} - y^{2} = 9 is$
$\begin{array}{l} (x^{2} - y^{2} - 9) (1^{2} - 0^{2} - 9) = (x \cdot 1 - y \cdot 0 - 9)^{2} (Using {SS}_{1} = T^{2}) \\ or (x^{2} - y^{2} - 9) (- 8) = (x - 9)^{2} \\ or - 8 x^{2} + 8 y^{2} + 72 = x^{2} - 18 x + 81 \\ or 9 x^{2} - 8 y^{2} - 18 x + 9 = 0 \end{array}$

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