CONIC SECTIONS - THE ELLIPSE

Example 1:

Find the coordinates of the center, the vertices, and foci of the ellipse given by .

This equation is almost in standard form. All we have to do is the following:

where h = 4, k = 3, a = 3 (from larger denominator), and b = 2

Now, we can find the requested information as follows:

Coordinates of the Center (h, k)

(4, 3)

Coordinates of the Vertices (h + a, k) and (h a, k)

(4 + 3, 3) = (1, 3)

and (4 3, 3) = (7, 3)

Coordinates of the Foci (h + c, k) and (h c, k), where

Since a = 3 and b = 2 , then .

and

Example 2:

Find the coordinates of the center, vertices, and foci of the ellipse given by .

First, we will convert the equation of the ellipse to standard form.

It is either or , where

First, we will divide both sides of the equation by 16 to get

 

We can now write the standard form of the given elliptic equation as follows:

where h = 1, k = 2, a = 4 (from larger denominator), and b = 2

Now, we can find the requested information as follows:

Coordinates of the Center (h, k)

(1, 2)

Coordinates of the Vertices (h, k + a) and (h, k a)

(1, 2 + 4) and (1, 2 4)

(1, 2) and (1, 6)

Coordinates of the Foci (h, k + c) and (h, k c), where

Since a = 4 and b = 2, then .

and

Example 3:

Find the coordinates of the center, vertices, and foci of the ellipse given by

.

First, we will convert the equation of the ellipse to standard form , where .

NOTE: The center (h, k) of the given ellipse is at (0, 0) because there is only an x²-term and a y²-term. Therefore, we can write

where h = 0, k = 0, a = 5 (from larger denominator), and b = 4

Now, we can find the requested information as follows:

Coordinates of the Center (h, k)

(0, 0)

Coordinates of the Vertices (h, k + a) and (h, k a)

(0, 0 + 5)and (0, 0 5)

(0, 5) and (0, 5)

Coordinates of the Foci (h, k + c) and (h, k c), where

Since a = 5 and b = 4, then .

(0, 0 + 3)and (0, 0 3)

(0, 3) and (0, 3)

Example 4:

Find the coordinates of the center, vertices, and foci of the ellipse given by .

First, we will convert the equation of the ellipse to standard form.

It is either or , where

First, we will divide both sides of the equation by 36 to get

We can now write the standard form of the given elliptic equation as follows:

where h = 0, k = 0, a = 3 (from larger denominator), and b = 2

Now, we can find the requested information as follows:

Coordinates of the Center (h, k)

(0, 0)

Coordinates of the Vertices (h, k + a) and (h, k a)

(0, 0 + 3) and (0, 0 3)

(0, 3) and (0, 3)

Coordinates of the Foci (h, k + c) and (h, k c), where

Since a = 3 and b = 2, then .

and

and

Example 5:

Match the names hyperbola, ellipse, parabola, and circle to the following conic equations.

a.

We must recognize this as a hyperbola because of its general characteristics. There is a squared x and y-term and one of them is negative. It is actually a hyperbola with center at the origin.

b.

We must recognize this as an ellipse because of its general characteristics. There is a squared x and y-term and both are positive. Also, the squared terms have different denominators.

c.

We must recognize this as an ellipse because of its general characteristics. There is a squared x and y-term and both are positive. Also, the squared terms have different coefficients (or denominators).

e.

We must recognize this as a circle because of its general characteristics. There is a squared x and y-term and both are positive. Also, the squared terms have the same denominators.

f.

We must recognize this as a circle because of its general characteristics. There is a squared x and y-term and both are positive. Also, the squared terms have the same coefficients (or denominators), namely 1.

d.

We must recognize this as the standard form of a parabola because of its general characteristics. There is only one squared term, namely y².