9.2 Ellipses
Given two points F1 and F2 in the plane lying a distance 2c apart and given a distance 2a > 2c, the set of points whose distances to F1 and F2 respectively sum to 2a is an ellipse. F1 and F2 are called the foci of the ellipse.
Exercise 9.2.1
Get a compass and a blank sheet of paper. On the sheet of paper, mark two points F1 and F2 and draw a dotted line through them. Construct a dotted perpendicular line to F1F2 through the midpoint of that segment. Label the midpoint as the center. Place the point of the compass at the center and open the compass to a fixed radius greater than the distance from the center to the foci. Mark points V1 and V2 where the pencil of the compass crosses the line that passes through F1 and F2. Without changing the radius on the compass, move the point of the compass to F1 and mark points V3 and V4 on the perpendicular that you constructed earlier. Then the points V1, V2, V3, V4 are the vertices of an ellipse. Draw a closed curve through the four points to represent the ellipse. The points F1 and F2 are called the foci of the ellipse. The line through the foci is called the major axis, and the line through the center and perpendicular to the major axis is called the minor axis.
Exercise 9.2.2
Using the ellipse that you drew in Exercise 9.2.1, denote the distance from the center to V1 and V2 as a. Find the sum of the distances from V1 to F1 and V1 to F2 in terms of a. Do the same for the sum of the distances from V2 to F1 and V2 to F2. Recall how the points V3 and V4 were constructed. Find the sum of the distances from V3 to F1 and V3 to F2 in terms of a. Do the same for the sum of the distances from V4 to F1 and V4 to F2.
Exercise 9.2.3
Let a denote the distance from the center to the major vertices V1 or V2 and let c denote the distance from the center to the foci F1 or F2. Let b denote the distance from the center to the minor vertices V3 or V4. Find an equation relating the relationship between a, b and c.
We will consider only ellipses with either a vertical or a horizontal major axis. Let us begin by considering an ellipse with center at the origin and foci ( ± c, 0 ) on the x-axis. Let ( ± a, 0 ) denote the major vertices and ( 0, ± b ) the minor vertices. The equation of this ‘horizontal’ ellipse is
If this ellipse is shifted h units horizontally and k units vertically, the resulting ellipse will have equation
In either case c2 = a2 – b2.
Exercise 9.2.4
Find the equation of the ellipse with center ( 0, 0 ), focus ( -3, 0 ) and minor vertex ( 0, 4 ).
Exercise 9.2.5
Find the equation of the ellipse with center ( -2, 3 ), focus ( -3, 3 ) and major vertex ( 5, 3 ).
Next, let us consider an ellipse with center at the origin and foci ( 0, ± c ) on the x-axis. Let ( 0, ± a ) denote the major vertices and (± b, 0 ) the minor vertices. The equation of this ‘vertical’ ellipse is
If this ellipse is shifted h units horizontally and k units vertically, the resulting ellipse will have equation
Exercise 9.2.6
Find the equation of the ellipse with center ( 0, 0 ), focus ( 0, 2 ) and major vertex ( 0, -4 ).
Exercise 9.2.7
Find the equation of the ellipse with center ( 1, 3 ), major vertex ( 1, 0 ) and minor vertex ( 2, 3 ).
Be able to complete the square to put the equation of an ellipse in standard form. Be able to find the center, foci and vertices of an ellipse, given its equation.
For example, consider the equation:
x2 + 9 y2 – 2 x + 36 y + 28 = 0
To put this in standard form, we first separate the variables
x2 – 2 x + 9 y2 + 36 y = - 28
Then we complete the squares on the two variables
(x2 – 2 x + 1 ) + 9 (y2 + 4 y + 4 ) = - 28 + 1 + 9 ( 4 )
( x – 1 ) 2 + 9 ( y + 2 ) 2 = 9
Then we divide by 9 to get
Thus, the ellipse is horizontal, a = 3, b = 1,
and 
.
The center is ( 1, -2 ), the major vertices are ( 1 ± 3, -2 ) or ( 4, -2 ) and ( -2, -2 ). The minor vertices are ( 1, -2 ± 1 ) or
( 1, -3 ) and ( 1, -1 ).
The ellipse may be sketched through these four vertices. The foci are 
.
Exercise 9.2.8
Find the center, vertices and foci of the ellipse with equation
25 x2 + 9 y2 – 50 x + 18 y - 191 = 0