In Exercises 37-50, graph each ellipse and give the location of its foci. 36(x+4)^2+(y+3)^2=36

Name: Problem 50, Chapter 10: Algebra and Trigonometry
Uploaded: 2020-06-11T08:11:15-08:00
Duration: 1 min 55 s
Channel: Ankit Gupta
Description: In Exercises 37-50, graph each ellipse and give the location of its foci. 36(x+4)^2+(y+3)^2=36

step 1 Here we have given that 36 x plus 4 whole square plus y plus 3 whole square is equal to 36 now w

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

Key Concepts

Ellipse

An ellipse is a set of points in a plane such that the sum of the distances from any point on the ellipse to its two foci is constant. This fundamental definition leads to many geometric properties, including symmetry about its center and the characteristic shape that is longer in one direction than the other.

Standard Form of an Ellipse

The standard form of the ellipse equation is obtained by algebraically manipulating a given quadratic equation so that it fits the form (x-h)²/a² + (y-k)²/b² = 1, where (h, k) represents the center of the ellipse and a and b are the lengths of the semi-axes. This form is crucial for easily identifying the geometric features of the ellipse.

Center and Axes

Once the equation is in standard form, the center of the ellipse (h, k) is immediately apparent. The values a and b correspond to the lengths of the semi-major and semi-minor axes respectively. Determining whether the ellipse is oriented horizontally or vertically depends on which of a or b is greater, guiding the proper interpretation and graphing of the ellipse.

Foci Calculation

The foci of an ellipse are two special points located along the major axis. The distance from the center to each focus, denoted by c, is found using the relationship c² = a² - b² for an ellipse where a > b. Knowing the center, the major axis orientation, and the value of c, one can precisely locate the foci, which are critical in the geometric definition and properties of the ellipse.

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