In Exercises 37-50, graph each ellipse and give the location of its foci. ((x-4)^2)/(4)+(y^2)/(2

step 1 Here we have given that x minus 4 whole square divided by 4 plus y square divided by 25 is equal

In Exercises 37-50, graph each ellipse and give the location...

Key Concepts

Ellipse

An ellipse is a type of conic section that consists of all points in a plane for which the sum of the distances from two fixed points (the foci) is constant. It is a closed curve that is symmetric about both its major and minor axes.

Standard Form of an Ellipse

The standard form of an ellipse is given by either (x-h)²/a² + (y-k)²/b² = 1 or (x-h)²/b² + (y-k)²/a² = 1, where (h, k) is the center. The larger denominator corresponds to the square of the semi-major axis, and the smaller corresponds to the square of the semi-minor axis. This form facilitates the identification of the ellipse's geometric properties.

Foci of an Ellipse

The foci of an ellipse are two key points located along the major axis, and their positions are determined by the relationship c² = a² - b², where 'a' is the semi-major axis and 'b' is the semi-minor axis. These points are essential in defining the ellipse since the sum of the distances from any point on the ellipse to the two foci is constant.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, drawing its major and minor axes, and then sketching the curve based on the distances defined by the semi-major (a) and semi-minor (b) axes. Identifying the foci also helps in understanding the spatial relationship within the ellipse and can be used to verify the correctness of the graph.

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