Equation of Ellipse - Problems

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This is a tutorial with detailed solutions to problems related to the ellipse equation. For an interactive tutorial on the equation of the ellipse Go here.

Review

An ellipse with center at the origin (0,0), is the graph of                        

with a > b > 0

The length of the major axis is 2a, and the length of the minor axis is 2b. The two foci (foci is the plural of focus) are at (± c , 0) or at (0 , ± c), where c² = a² - b².

Problem 1:  Given the following equation

                                        9x² + 4y² = 36

a) Find the x and y intercepts of the graph of the equation.

b) Find the coordinates of the foci.

c) Find the length of the major and minor axes.

d) Sketch the graph of the equation.

Solution to Example 1

a) We first write the given equation in standard form by dividing both sides of the equation by 36

                                        9x² / 36 + 4y² / 36 = 1

                                        x² / 4 + y² / 9 = 1

                                        x² / 2² + y² / 3² = 1

We now identify the equation obtained with one of the standard equation in the review above and we can say that the given equation is that of an ellipse with a = 3 and b = 2 (NOTE: a >b) .

Set y = 0 in the equation obtained and find the x intercepts.

                                        x² / 2²  = 1

Solve for x.

                                        x²  = 2²

                                        x = ± 2

Set x = 0 in the equation obtained and find the y intercepts.

                                     y² / 3² = 1

Solve for y.

                                       y²  = 3²

                                        y = ± 3

b) We need to find c first.

                                     c² = a² - b²

a and b were found in part a).

                                     c² = 3² - 2²

                                     c² = 5

Solve for c.

                                    c = ± (5)^1/2

The foci are    F₁ (0 , (5)^1/2) and  F₂ (0 , -(5)^1/2)

c) The major axis length  is given by  2 a = 6.

 The minor axis length  is given by  2 b = 4.

d) Locate the x and y intercepts, find extra points if needed and sketch.

Matched Problem:  Given the following equation

                                        4x² + 9y² = 36

a) Find the x and y intercepts of the graph of the equation.

b) Find the coordinates of the foci.

c) Find the length of the major and minor axes.

d) Sketch the graph of the equation.