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.

검증된 단계별 안내

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 = r^{2}

, 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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주요 개념

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

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.

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.

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.