sample 8 : Equation of Ellipse

College algebra problems on the equation of the ellipse are presented. More Problems on ellipses with detailed solutions are included in this site. The solutions to the questions below are at the bottom of the page.

Find

a) the major axis and the minor axis of the ellipse and their lengths,

b) the vertices of the ellipse,

c) and the foci of this ellipse.

a) Find the equation of part of the graph of the given ellipse that is to the left of the y axis.

b) Find the equation of part of the graph of the given ellipse that is below the x axis.

Find

a) its center,

a) its major and minor axes and their lengths,

b) its vertices,

c) and the foci.

Find the center, the major and minor axes and their lengths of this ellipse.

Find

a) the center of the ellipse,

b) its major and minor axes and their lengths,

c) its vertices,

d) and its foci.

The above equation may be written in the form x

with a = 3/5 and b = 1/3. The major axis is the x axis and its length is equal to 2a = 6/5 = 1.2

a) Divide all terms of the equation by 32 to obtain

The above equation may be written as follows

with a = 4 and b = 2 and a > b. Hence the major axis is the y axis and the minor axis is the x axis. The length of the major axis = 2a = 8 and the length of the minor axis = 2b = 4

b) The vertices are on the major axis at the points (0 , a) = (0 , 4) and (0 , -a) = (0 , -4)

c) The foci are on the major axis at the points (0 , c) and (0 , -c) such that

c

Hence the foci are at the points (0 , 2√3) and (0 , -2√3)

a is the distance from the center of the ellipse to the a vertex and is equal to 6. c is the distance from the center of the ellipse to a focus and is equal to 4. Also a, b and c are related as follows

b

b = 2√5

The equation of the ellipse is given by

The equation of the ellipse is given by

Length of major axis is 10 hence a = 5 and the equation may be written as follows

We now use the fact that the point (3 , 16/5) lies on the graph of the ellipse to find b

Solve the above for b to find b = 4 and write the equation as follows

a) Solve the above equation for x and select the solution for which x is positive

x = √(1 - 3y

b) Solve the ellipse equation for y and select the solution for which y is negative.

y = - √(1/3 - (1/3)x

a) Ellipse with center at (h , k) = (1 , -4) with a = 4 and b = 3.

a) its major axis is the line x = 1, and its minor is the line y = -4. length of major axis = 2a = 8 , length of minor axis = 2b = 6

b) vertices at: (1 , -4 + 4) = (1 , 0) and (1 , -4 - 4) = (1 , -8)

c) c = √(a

Foci at: (1 , -4 + √7) and (1 , -4 - √7)

The center of the ellipse is the midpoint of the two foci and is at (2 , 0).

c is the length from one foci to the center, hence c = 2.

length of minor axis 2 = 2b hence b = 1

a

Since the foci are on the x axis, the major axis of the ellipse is the x axis.

Equation of ellipse:

The parametric equations can be written as follows:

x / 6 = sin(t) and y / 4 = cos(t)

Square both sides of the two equations: (x / 6)

Use the fact that sin

x

a = 6 , b = 4, c = √(36 - 16) = 2√5

major axis: x axis , length = 12

minor axis: y axis , length = 8

Problems on ellipses

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