How do you write an equation for the hyperbola with center (0,0), vertex (-2,0) and focus (4,0)?

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How do you write an equation for the hyperbola with center (0,0), vertex (-2,0) and focus (4,0)?

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Answer 1 · 2016-10-30T18:30:32

Please see the explanation.

Explanation:

The given, center, vertex, and focus share the same y coordinate, 0, ,therefore, the standard form for the equation of this type of hyperbola is the one corresponding to the Horizontal Transverse Axis type:

$(x - h)^2/a^2 - (y - k)^2/b^2 = 1$

where $(h, k)$ is the center, $a$ is distance from the center to the vertex, and $b$ affects the distance, $c$ , from the center to the focus as determined by the equation $c^2 = a^2 + b^2$ .

Substitute the center $(0, 0)$ into the standard form:

$(x - 0)^2/a^2 - (y - 0)^2/b^2 = 1$

We know that $a = 2$ , because one of the vertices a distance of 2 from the center. Substitute 2 for a into the equation:

$(x - 0)^2/2^2 - (y - 0)^2/b^2 = 1$

Observing that the focus is a distance of 4 from the center we set $c^2 = 16$ in the equation $c^2 = a^2 + b^2$ :

$16 = a^2 + b^2$

Substitute 4 for $a^2$

$16 = 4 + b^2$

Solve for b:

$b= sqrt(12)$

Substitute $sqrt(12)$ for b into the equation:

$(x - 0)^2/2^2 - (y - 0)^2/(sqrt(12))^2 = 1$

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How do you write an equation for the hyperbola with center (0,0), vertex (-2,0) and focus (4,0)?

1 Answer

Explanation:

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