39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.

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Identify the type of conic section: Since the graph shows a hyperbola centered at the origin with vertices and foci along the y-axis, it is a vertical hyperbola.

Recall the standard form of the equation for a hyperbola centered at the origin with a vertical transverse axis: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

Determine the value of \(a\): The vertices are at \((0, 6)\) and \((0, -6)\), so the distance from the center to each vertex is \(a = 6\), which means \(a^2 = 36\).

Determine the value of \(c\): The foci are at \((0, 10)\) and \((0, -10)\), so the distance from the center to each focus is \(c = 10\), which means \(c^2 = 100\).

Use the relationship between \(a\), \(b\), and \(c\) for hyperbolas: \(c^2 = a^2 + b^2\). Substitute the known values to solve for \(b^2\): \(100 = 36 + b^2\), then \(b^2 = 100 - 36\).

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주요 개념

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Standard Equation of a Hyperbola Centered at the Origin

A hyperbola centered at the origin with a vertical transverse axis has the equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). Here, \(a\) is the distance from the center to each vertex along the y-axis, and \(b\) relates to the distance along the conjugate axis. This form is essential to write the equation based on given vertices and foci.

Relationship Between Vertices, Foci, and Parameters \(a\), \(b\), and \(c\)

In a hyperbola, \(a\) is the distance from the center to each vertex, and \(c\) is the distance from the center to each focus. These satisfy the equation \( c^2 = a^2 + b^2 \). Knowing \(a\) and \(c\) allows calculation of \(b\), which is necessary to complete the hyperbola's equation.

Graph Interpretation and Coordinate Identification

Analyzing the graph helps identify key points such as vertices and foci coordinates. For this hyperbola, vertices at (0,6) and (0,-6) give \(a=6\), and foci at (0,10) and (0,-10) give \(c=10\). These values are critical inputs for forming the hyperbola's equation.