6-3 Ellipses

I am going to explain how to do ellipses. There are seven steps to working an ellipse.

Standard form of an ellipse is as follows:

Put an ellipse into standard form by dividing or completing the square.

1. 1. .Define the major axis (The variable with the largest denominator is major)

2. 2Define the minor axis (The variable with the smaller denominator is minor)

3. 3. Find the vertex. The vertex is the square root of the major axis (larger denominator). It is put into point form. The answer goes into the slot of the major axis.

4. 4.Find the other intercept. The other intercept is the square root of the minor axis (smaller denominator). It is put into point form and the answer goes into the slot of the minor axis.

5. 5 Find the length of the major axis. This is found by multiplying the square root of the major axis by two.

6. 6.. Find the length of the minor axis. This is found by multiplying the square root of the major axis by two.

7. 7. Find the focus. (The focus is ALWAYS on the major axis.) It is in point form. The equation for finding the focus, vertex, or other intercept is smallest denom=largest denom-focus^2.

EXAMPLE

Sketch the ellipse.

X^2/25+y^2/169=1

1. . Y is major.

Again, the larger denominator is the major axis.

2. 2. X is minor.

The smaller denominator is the minor axis.

3. 3. Vertex=square root of 169=+/-13 (0,13) (0,-13)

The vertex will always be the square root of the major axis put into point form.

4. 4. Other intercept=square root of 25=+/-5 (5,0) (-5,0)

The other intercept is always the square root of the minor axis put into point form.

5. 5. 2*square root of 169=26

The length of the major axis (in this case y) is found by multiplying 2*the square root of the larger denom (in this case 169) which would make it 2*13=26.

6. 6. 2*square root of 25=10

The length of the minor axis (in this case x) is found by multiplying 2*the square root of the smaller denom (in this case 25) which would make it 2*5=10.

7. 7. 25=169-f^2

f^2=144

f=+/-12 (0,12) (0,-12)

The focus is found using the equation smaller denom=larger denom-f^2. You first subtract 169 from both sides. Then you square root both sides to get +/-12 and put it into point form. THE FOCUS IS ALWAYS ON THE MAJOR AXIS.

The graph would look like this.

That’s an ellipse.

--Sarah