You and your friends are playing Name the Conic Section. Your friend pulls a card with the equation written on it. What type of conic section is represented by the equation?
Guidance
Another way to classify a conic section when it is in the general form is to use the discriminant, like from the Quadratic Formula. The discriminant is what is underneath the radical, , and we can use this to determine if the conic is a parabola, circle, ellipse, or hyperbola. If the general form of the equation is , where , then the discriminant will be .
Use the table below:
and or  Parabola 

and  Circle 
and  Ellipse 
Hyperbola 
Example A
Use the discriminant to determine the type conic: .
Solution: , , and
This is a hyperbola.
Example B
Use the discriminant to determine the type of conic:
Solution: , ,
Because and the discriminant is less than zero, this is a circle.
Example C
Use the discriminant to determine the type of conic. Then, change the equation into standard form to verify your answer. Find the center or vertex, if it is a parabola.
Solution: This is a circle.
The center is .
Intro Problem Revisit First we need to rewrite the equation is standard form.
Now we can use the discriminant to find the type of conic section represented by the equation.
Because and the discriminant is less than zero, this equation represents an ellipse.
Guided Practice
Use the discriminant to determine the type of conic.
1.
2.
3. Use the discriminant to determine the type of conic. Then, change the equation into standard form to verify your answer. Find the center or vertex, if it is a parabola.
Answers
1. , this is an ellipse.
2. , this is a parabola.
3. , this is a hyperbola. Changing it to standard form, we have:
Usually, we write the negative term second, so the equation is . The center is .
Vocabulary
 Discriminant
 When referring to the general seconddegree equation, , the discriminant is and it determines the type of conic the equation represents.
Practice
Use the discriminant to determine the type of conic each equation represents.
Match the equation with the correct graph.
Use the discriminant to determine the type of conic. Then, change the equation into standard form to verify your answer. Find the center or vertex, if it is a parabola.

Find the Area of an Ellipse
Graph
and find is area.
 Then, graph and on the same axes.
 Do these ellipses have the same area? Why or why not?
 If the equation of the area of a circle is , what do you think the area of an ellipse is? Use and as in the standard form, .
 Find the areas of the ellipses from part a. Are the areas more or less than the area of the circle? Why or why not?