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

A hyperbola with vertices (±4, 0) and foci (±6, 0)

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Identify the type of conic: Since the vertices and foci lie on the x-axis, the hyperbola opens left and right, so its standard form is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Determine the value of \(a\): The vertices are at \((\pm 4, 0)\), so \(a = 4\) and therefore \(a^2 = 16\).

Determine the value of \(c\): The foci are at \((\pm 6, 0)\), so \(c = 6\) and therefore \(c^2 = 36\).

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

Write the equation of the hyperbola using the values of \(a^2\) and \(b^2\): \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Standard Form of a Hyperbola Centered at the Origin

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

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

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

Interpreting Coordinates of Vertices and Foci

Vertices and foci coordinates provide direct information about the hyperbola's orientation and size. Vertices at (±4, 0) indicate a horizontal transverse axis with \(a=4\), and foci at (±6, 0) give \(c=6\). This helps determine the parameters needed to write the hyperbola's equation.

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. A hyperbola with vertices (±4, 0) and foci (±6, 0)