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Question
Find the centre, eccentricity, foci and directrice of the hyperbola .
x2 − 3y2 − 2x = 8.
Solution
The equation x2 − 3y2 − 2x = 8 can be simplified in the following manner:
\[( x^2 - 2x + 1) - 3 y^2 = 8 + 1\]
\[ \Rightarrow \left( x - 1 \right)^2 - 3 y^2 = 9\]
\[ \Rightarrow \frac{(x - 1 )^2}{9} - \frac{y^2}{3} = 1\]
Thus, the centre is \[\left( 1, 0 \right)\] .
Eccentricity of the hyperbola = \[\frac{\sqrt{a^2 + b^2}}{a} = \frac{\sqrt{9 + 3}}{3} = \frac{2\sqrt{3}}{3}\]
Foci = \[\left( 1 \pm 2\sqrt{3}, 0 \right)\]
Equation of the directrices:
\[x - 1 = \pm \frac{a}{e}\]
\[ \Rightarrow x = 1 \pm \frac{2\sqrt{3}}{3}\]
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