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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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 \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\).

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주요 개념

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Standard Form of a Hyperbola

The standard form of a hyperbola equation depends on its orientation. For a vertical transverse axis centered at the origin, the equation is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to each vertex, and \(b\) relates to the conjugate axis. Recognizing the correct form is essential to write the equation properly.

Relationship Between Vertices, Foci, and Parameters

Vertices and foci determine the values of \(a\) and \(c\) respectively, where \(a\) is the distance from the center to a vertex, and \(c\) is the distance from the center to a focus. For hyperbolas, these satisfy \( c^2 = a^2 + b^2 \). Knowing \(a\) and \(c\) allows calculation of \(b\), which completes the equation.

Center and Orientation of the Hyperbola

The center of the hyperbola is the midpoint between its vertices and foci. In this problem, both vertices and foci lie on the y-axis, indicating a vertical transverse axis and a center at the origin (0,0). Identifying the center and orientation guides the selection of the correct standard form and variable placement.