Finding the Standard Equation of an Ellipse In Exercises...

Finding the Standard Equation of an Ellipse In Exercises $31-36,$ find the standard form of the equation of the ellipse with the given characteristics.
Center: $(0,0)$
Major axis: horizontal
Points on the ellipse:
$(3,1),(4,0)$

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse’s equation is given by ((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) = 1, where (h, k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively. This form provides a clear geometric interpretation of the ellipse in relation to its size and position in the coordinate plane.

Horizontal and Vertical Orientation

The orientation of an ellipse depends on the relative values of a and b. If a is greater than b, the ellipse has a horizontal major axis, meaning it is stretched more along the x-axis; if b is greater than a, the ellipse has a vertical major axis, meaning it is stretched more along the y-axis. This concept is crucial for understanding the directional properties of the ellipse.

Using Known Points to Determine Parameters

When certain points are known to lie on an ellipse, those points can be substituted into the ellipse’s standard equation to create equations involving a and b. Solving these equations helps in determining the specific values of the semi-major and semi-minor axes, thus fully defining the ellipse’s unique shape.

Center of an Ellipse

The center of an ellipse is the point that is equidistant from all the vertices along the major and minor axes. It serves as the reference point for transforming the general equation of the ellipse, making it easier to analyze and graph the ellipse in a coordinate system.

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