Find the standard form of the equation for an ellipse satisfying the given conditions. Center (0

step 1 In this question, we are going to write the standard form equation of an ellipse that satisfies

Find the standard form of the equation for an ellipse...

Key Concepts

Semi-Axes and Diameter Conversion

In any ellipse, the semi-major and semi-minor axes are obtained by halving the respective lengths of the major and minor diameters. These semi-axis lengths (denoted as a and b) are the key values used in the standard equation to represent the distances from the center to the boundary of the ellipse along the principal directions.

Major and Minor Axes

The major axis is the longest line segment that passes through the center of the ellipse, while the minor axis is the shortest, perpendicular to the major axis at the center. The lengths of these axes determine the overall dimensions of the ellipse and indicate its orientation in the plane.

Ellipse

An ellipse is a type of conic section defined as the set of all points in a plane for which the sum of the distances from two fixed points (the foci) is constant. It is characterized by its symmetry about two perpendicular axes and its oval shape, making it a fundamental example used in the study of conic sections.

Standard Form Equation of an Ellipse

The standard form of an ellipse's equation is a concise way to describe its shape and position in the coordinate plane. When centered at (h, k), the equation is written as ((x-h)²)/(a²) + ((y-k)²)/(b²) = 1 for a horizontal major axis or with the roles of a and b swapped for a vertical major axis. This form clearly displays the lengths of the semi-axes and the orientation of the ellipse.

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