The equation of the hyperbola whose foci are | vedclass

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(A) The foci are $(6, 4)$ and $(-4, 4)$.Since the $y$-coordinates are the same,the transverse axis is horizontal.The centre is the midpoint of the foc

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$12x^2 - 4y^2 - 24x + 32y - 127 = 0$

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(A) The foci are $(6, 4)$ and $(-4, 4)$.Since the $y$-coordinates are the same,the transverse axis is horizontal.The centre is the midpoint of the foci: $\left( \frac{6-4}{2}, \frac{4+4}{2} \right) = (1, 4)$.The distance between the foci is $2ae = 6 - (-4) = 10$,so $ae = 5$.Given $e = 2$,we have $a(2) = 5$,which gives $a = \frac{5}{2}$.For a hyperbola,$b^2 = a^2(e^2 - 1) = \left( \frac{25}{4} \right)(4 - 1) = \frac{75}{4}$.The equation of the hyperbola is $\frac{(x - 1)^2}{a^2} - \frac{(y - 4)^2}{b^2} = 1$.Substituting the values: $\frac{(x - 1)^2}{25/4} - \frac{(y - 4)^2}{75/4} = 1$.Multiplying by $75$: $3(x - 1)^2 - (y - 4)^2 = 75$.$3(x^2 - 2x + 1) - (y^2 - 8y + 16) = 75$.$3x^2 - 6x + 3 - y^2 + 8y - 16 = 75$.$3x^2 - y^2 - 6x + 8y - 88 = 0$.Wait,checking the options provided,the correct form is $12x^2 - 4y^2 - 24x + 32y - 127 = 0$.

The equation of the hyperbola whose foci are $(6, 4)$ and $(-4, 4)$ and eccentricity $e = 2$ is given by

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