In Exercises 37-50, graph each ellipse and give the location of its foci. ((x+3)^2)/(9)+(y-2)^2=

step 1 We have given here that x plus 3 whole square divided by 9 plus y minus 2 whole square is equal

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

Key Concepts

Ellipse Equation in Standard Form

An ellipse can be expressed in a standard form equation that clearly shows its geometric properties. This standard form, ((x-h)²)/(a²) + ((y-k)²)/(b²) = 1, identifies the center (h, k) of the ellipse, the lengths of the semi-major axis (a) and semi-minor axis (b), and indicates the orientation of the ellipse depending on whether a > b or b > a.

Identifying the Center and Axes

The center of an ellipse is found by looking at the shifts in the x and y terms from the standard form equation. The denominators of the equation provide the squares of the lengths of the semi-axes. Specifically, a² corresponds to the square of the length of the semi-major axis and b² to that of the semi-minor axis, which helps in understanding the size and shape of the ellipse.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, then using the lengths of the semi-major and semi-minor axes to determine and plot the vertices and co-vertices. This process provides a clear visual representation of the ellipse, including its orientation and spread in the coordinate plane.

Calculating the Foci

The foci of an ellipse are two important points located along the major axis. Their positions are determined by the relationship c = ?(a² - b²), where a is the length of the semi-major axis and b the length of the semi-minor axis. These foci are essential in defining the ellipse, as any point on the ellipse has the constant sum of distances to these fixed points.

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