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प्रश्न
Find the centre, eccentricity, foci and directrice of the hyperbola .
16x2 − 9y2 + 32x + 36y − 164 = 0
उत्तर
The equation \[16 x^2 - 9 y^2 + 32x + 36y - 164 = 0\] can be simplified in the following way:
\[16( x^2 + 2x) - 9\left( y^2 - 4y \right) = 164\]
\[ \Rightarrow 16( x^2 + 2x + 1) - 9\left( y^2 - 4y + 4 \right) = 164 + 16 - 36\]
\[ \Rightarrow 16(x + 1 )^2 - 9 \left( y - 2 \right)^2 = 144\]
\[ \Rightarrow \frac{(x + 1 )^2}{9} - \frac{(y - 2 )^2}{16} = 1\]
Thus, the centre is \[\left( - 1, 2 \right)\].
Eccentricity of the hyperbola = \[\frac{\sqrt{a^2 + b^2}}{a} = \frac{\sqrt{9 + 16}}{3} = \frac{5}{3}\]
Foci = \[\left( - 1 \pm 5, 2 \right) = \left( - 6, 2 \right), \left( 4, 2 \right)\]
Equation of the directrices:
\[x + 1 = \pm \frac{a}{e}\]
\[ \Rightarrow x = \pm \frac{3 \times 3}{5} - 1\]
\[ \Rightarrow x = \pm \frac{9}{5} - 1\]
\[ \Rightarrow 5x - 4 = 0 \text { or } 5x + 14 = 0\]
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