In Exercises 37-50, graph each ellipse and give the location of its foci. ((x-4)^2)/(9)+((y+2)^2

step 1 Here we have given that x minus 4 whole square divided by 9 plus y plus 2 whole square divided b

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

Key Concepts

Foci of an Ellipse

The foci of an ellipse are two special points along its major axis that satisfy the definition of the ellipse, where the sum of the distances from any point on the ellipse to the foci is constant. The distance from the center to each focus, denoted by c, is calculated using the relationship c² = a² - b² (with a corresponding to the semi-major axis length). The foci are then located along the major axis, symmetrically about the center.

Ellipse

An ellipse is a set of all points in a plane such that the sum of the distances from two fixed points (foci) is constant. It is a common conic section and has standard geometric properties including a center, vertices, and axes that determine its shape and orientation.

Standard Form of an Ellipse

The standard form of an ellipse's equation is ((x - h)²)/a² + ((y - k)²)/b² = 1, where (h, k) is the center, and a and b are the lengths of the semi-major and semi-minor axes respectively. This form helps in easily identifying the key features of the ellipse such as its center, axes, vertices, and orientation.

Orientation and Axes

In the standard form, the denominator with the larger value corresponds to the square of the semi-major axis. If the larger denominator is under the (x - h)² term, the ellipse is horizontally oriented; if it is under the (y - k)² term, the ellipse is vertically oriented. This distinction is important for graphing the ellipse correctly and for determining the locations of its vertices and foci.

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