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

Verified step by step guidance

1. Identify the type of conic section: Since the graph shows two branches opening left and right with vertices at (±4, 0) and foci at (±5, 0), this is a hyperbola centered at the origin with a horizontal transverse axis.
2. Recall the standard form of the equation of a hyperbola centered at the origin with a horizontal transverse axis: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
3. Determine the value of \(a\): The vertices are at (±4, 0), so \(a = 4\), which means \(a^2 = 16\).
4. Determine the value of \(c\): The foci are at (±5, 0), so \(c = 5\), which means \(c^2 = 25\).
5. Use the relationship between \(a\), \(b\), and \(c\) for hyperbolas: \(c^2 = a^2 + b^2\). Substitute the known values to solve for \(b^2\): \$25 = 16 + b^2\(, then solve for \)b^2$.

Key Concepts

Standard Equation 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 distance along the conjugate axis. Understanding this form is essential to write the equation from 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 the coordinates of vertices and foci allows calculation of \(a\), \(c\), and subsequently \(b\), which are needed to form the hyperbola's equation.

Graph Interpretation and Coordinate Geometry

Interpreting the graph involves identifying key points such as vertices and foci and their coordinates. This helps in determining the values of \(a\) and \(c\) directly from the graph. Accurate reading of these points is crucial for applying the hyperbola formulas correctly.