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Ch. 7 - Conic Sections
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 7 - Conic SectionsProblem 89
Chapter 8, Problem 89

The equation of the red ellipse in the figure shown is x^2/25 + y^2/9 =1Write the equation for each circle shown in the figure.
Graph showing a red ellipse and three circles in a coordinate plane.

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Identify the center of each circle. Since all circles are centered at the origin (0,0), their equations will be of the form x^2 + y^2 = r2, where r is the radius.
Determine the radius of each circle by observing where each circle intersects the x-axis or y-axis. The radius is the distance from the center to these points.
For the green circle, note the intersection points on the axes and use the distance from the origin to these points as the radius.
For the blue circle, similarly find the radius by measuring the distance from the origin to the points where the circle crosses the axes.
For the brown circle, find the radius by identifying the farthest points on the axes and use that distance as the radius to write the equation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation of an Ellipse

An ellipse centered at the origin has the equation x²/a² + y²/b² = 1, where a and b are the lengths of the semi-major and semi-minor axes. The given ellipse has a = 5 and b = 3, indicating its horizontal and vertical stretches. Understanding this form helps distinguish ellipses from circles.
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Foci and Vertices of an Ellipse

Equation of a Circle

A circle centered at the origin has the equation x² + y² = r², where r is the radius. Each circle in the figure can be described by identifying its radius from the graph and substituting it into this formula. Recognizing this standard form is essential for writing the equations of the circles.
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Circles in Standard Form

Graph Interpretation and Radius Measurement

Interpreting the graph involves identifying the radius of each circle by measuring the distance from the center (origin) to any point on the circle along the x or y axis. This measurement is crucial to formulating the correct equation for each circle, as the radius directly determines the equation's constant term.
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Graphs and Coordinates - Example
Related Practice
Textbook Question

Graph each semiellipse. y = -√16 - 4x²

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Textbook Question

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. {x=(y+2)2−1(x−2)2+(y+2)2=1\(\begin{cases}\)x = (y + 2)^2 - 1 \\(x - 2)^2 + (y + 2)^2 = 1\(\end{cases}\)

Textbook Question

Find the standard form of the equation of an ellipse with vertices at (0, -6) and (0, 6), passing through (2, 4).

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Textbook Question

Find the standard form of the equation of the hyperbola with vertices (5, −6) and (5, 6), passing through (0, 9).

Textbook Question

Solve by eliminating variables: