Textbook Question
31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.
A parabola symmetric about the y-axis that passes through the point (2, -6)
54
views
16. Parametric Equations & Polar Coordinates
Conic Sections
3:36 minutesProblem 12.4.41
Textbook Question
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)
Verified step by step guidance1
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 \(b^2\): \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
5:59
Graph Hyperbolas NOT at the Origin
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.
Recommended video:
5:22Foci and Vertices of Hyperbolas
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.
Recommended video:
5:30Foci and Vertices of an Ellipse
Related Videos
Related Practice