Question

Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (0, −3), (0, 3) ; vertices: (0, −1), (0, 1)

Accepted Answer

Identify the orientation of the hyperbola by examining the coordinates of the foci and vertices. Since both foci and vertices lie on the y-axis, the hyperbola opens vertically. Recall the standard form of a vertical hyperbola centered at the origin: $$\frac{x$$^{2}$$}{a$$^{2}$$} - \frac{y$$^{2}$$}{b$$^{2}$$} = -1 or $$equivalently $$\frac{y$$^{2}$$}{a$$^{2}$$} - \frac{x$$^{2}$$}{b$$^{2}$$} = 1. $$Since the hyperbola opens vertically, use $$\frac{y$$^{2}$$}{a$$^{2}$$} - \frac{x$$^{2}$$}{b$$^{2}$$} = 1. $$Determine the value of $$a$$ using the vertices. The vertices are at $$(0, \pm a)$$, so $$a is $$the distance from the center to a vertex. Here, $$a = 1. $$Determine the value of $$c$$ using the foci. The foci are at $$(0, \pm c)$$, so $$c is $$the distance from the center to a focus. Here, $$c = 3. $$Use the relationship between $$a$$, $$b$$, and $$c$$ for hyperbolas: $$c^{2}$$ = $$a^{2}$$ + $$b^{2}. $$Substitute $$a$$ and $$c to $$solve for $$b^{2}$$, then write the standard form equation $$\frac{y$$^{2}$$}{a$$^{2}$$} - \frac{x$$^{2}$$}{b$$^{2}$$} = 1.$$

Find the standard form of the equation of each hyperbola satisfying the given conditions. Foci: (0, −3), (0, 3) ; vertices: (0, −1), (0, 1)

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Key Concepts

Standard Form of a Hyperbola

Relationship Between Vertices, Foci, and Parameters

Center and Orientation of the Hyperbola