In Exercises 25-36, find the standard form of the equation of each ellipse satisfying the given

step 1 In this question we have given that fosai of this ellipse is minus 2 .0 and 2 .0 and y intercept

In Exercises 25-36, find the standard form of the equation...

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse’s equation expresses the relationship between the coordinates of its points and its key parameters. For an ellipse centered at the origin, the equation is either (x²)/(a²) + (y²)/(b²) = 1 or (x²)/(b²) + (y²)/(a²) = 1, where a and b are the lengths of the semi-axes. The larger denominator corresponds to the major axis, which determines the direction in which the ellipse is stretched.

Focal Relationship in an Ellipse

In any ellipse, the distance from the center to each focus, denoted by c, is related to the lengths of the semi-major and semi-minor axes by the equation c² = a² - b² (if the ellipse is oriented horizontally) or c² = b² - a² (if oriented vertically). This relationship is crucial for determining one of the parameters when the other two are known, and it encapsulates the geometric definition of an ellipse as the set of points for which the sum of the distances to the two foci is constant.

Intercepts on Coordinate Axes

Intercepts occur where the ellipse crosses the coordinate axes and are found by setting one variable to zero. For example, setting x = 0 in the standard form of the ellipse’s equation yields the y-intercepts, which can help determine the values of a or b. These intercepts provide vital information about the size and orientation of the ellipse and are often used to solve for unknown parameters in the ellipse's equation.

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