The equation of the red ellipse in the figure shown is (x^2)/(25)+(y^2)/(9)=1 Write the equation

step 1 Here we have given a diagram and the ellipse in red has a standard equation which is equals to x

The equation of the red ellipse in the figure shown is...

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Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse's equation is given by (x - h)²/a² + (y - k)²/b² = 1, where (h, k) represents the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes respectively. Understanding this form helps in identifying the properties of the ellipse, such as its orientation and dimensions, which is essential when comparing it to other conic sections like circles.

Standard Form of a Circle

The standard form of a circle's equation is written as (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is its radius. This formulation is fundamental in analytic geometry as it directly connects the algebraic representation of the circle to its geometric features, enabling us to determine its location and size on the coordinate plane.

Center and Radius in Conic Sections

A key concept in the study of circles and ellipses is the identification of the center and radius (or the semi-axes for ellipses). The center, given by (h, k), is the point of symmetry for these conic sections, and the radius for circles or the semi-axis lengths for ellipses determine the spread or size of the graph. Recognizing these parameters allows one to graph these shapes accurately and understand their relationships in a coordinate system.

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