The distance between the foci of a hyperbola | vedclass

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(C) Let the equation of the hyperbola be $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Given that the distance between the foci is double the distance betw

Vedclass · Accepted Answer

$3x^2 - y^2 = 9$

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(C) Let the equation of the hyperbola be $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Given that the distance between the foci is double the distance between its vertices: $2ae = 2(2a) \implies ae = 2a \implies e = 2$. We know that $e^2 = 1 + \frac{b^2}{a^2}$,so $4 = 1 + \frac{b^2}{a^2} \implies \frac{b^2}{a^2} = 3 \implies b^2 = 3a^2$. The length of the conjugate axis is $2b = 6$,which implies $b = 3$,so $b^2 = 9$. Substituting $b^2 = 9$ into $b^2 = 3a^2$,we get $9 = 3a^2 \implies a^2 = 3$. The equation of the hyperbola is $\frac{x^2}{3} - \frac{y^2}{9} = 1$,which simplifies to $3x^2 - y^2 = 9$.

The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is $6$. The equation of the hyperbola referred to its axes as axes of coordinates is

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