Identify a conic section from the characteristics of it equation. Identify a conic section from its general form equation.

1. 10.4 Conic Sections Summary
Chapter 10 Analytic Geometry

2. Concepts & Objectives
 Identify characteristics of each type of conic section
 Identify a conic section from its equation in general form
 Identifying the eccentricities of each type of conic
section

3. Conic General Form
 All of the conic sections we have been studying have
equations of the general form:
 Either A or C must be nonzero.
 There’s no B because that would be an xy term which
does weird things to the graph that’s beyond what we’re
studying.
    
2 2
0
Ax Cy Dx Ey F

4. Conic General Form (cont.)
 The special characteristics of the general equation of
each conic section are summarized below:
 For a full list, see the 10.4 Conic Sections Summary Chart
in Canvas,
    
2 2
0
Ax Cy Dx Ey F
Section Characteristic Example Eccentricity
Parabola Either A = 0 or C = 0,
but not both
e = 1
Circle A = C  0 e = 0
Ellipse A  C, AC > 0 0 < e < 1
Hyperbola AC < 0 e > 1
2
2
4 0
4 0
x y
x y y
  
   
  
2 2
16 0
x y
2 2
25 16 400 0
x y
  
  
2 2
1 0
x y

5. Identifying Conic Sections
 We can use these characteristics to help us generally
classify a conic general equation into a potential specific
conic section.
 Example: Classify the following equations.

 If we shift the y2 term to the left side, it becomes
negative – this is a hyperbola.
2 2
25 5
x y
 

6. Identifying Conic Sections
 We can use these characteristics to help us generally
classify a conic general equation into a potential specific
conic section.
 Example: Classify the following equations.
 Hyperbola

 The squared terms have positive, equal coefficients –
this is (potentially) a circle. We would have to
complete the square to find out what r is to classify it
further.
2 2
25 5
x y
 
2 2
8 10 41
x x y y
    

7. Identifying Conic Sections
 We can use these characteristics to help us generally
classify a conic general equation into a potential specific
conic section.
 Example: Classify the following equations.
 Hyperbola
 Circle (potentially)

 The coefficients of the squared terms are both positive
and different, so this should be an ellipse.
2 2
25 5
x y
 
2 2
8 10 41
x x y y
    
2 2
4 16 9 54 61
x x y y
    

8. Identifying Conic Sections
 We can use these characteristics to help us generally
classify a conic general equation into a potential specific
conic section.
 Example: Classify the following equations.
 Hyperbola
 Circle (potentially)
 Ellipse

 We have an x2 term, but no y2 term, so this is a
parabola.
2 2
25 5
x y
 
2 2
8 10 41
x x y y
    
2 2
4 16 9 54 61
x x y y
    
2
6 6 7 0
x y y
   

9. Identifying Conic Sections
 We can use these characteristics to help us generally
classify a conic general equation into a potential specific
conic section.
 Example: Classify the following equations.
 Hyperbola
 Circle (potentially)
 Ellipse
 Parabola
 Like the circle, the hyperbola and ellipse identification is
also dependent on making sure we don’t end up with
anything nonexistent. So how do we do that?
2 2
25 5
x y
 
2 2
8 10 41
x x y y
    
2 2
4 16 9 54 61
x x y y
    
2
6 6 7 0
x y y
   

10. Identifying Conic Sections (cont.)
 To turn the general form into something we can use, we
will have to complete the square.
 Once you have completed the process, you can identify
the resulting equation as an ellipse, a circle, a hyperbola,
or even a single point or nothing at all.

11. Identifying Conic Sections (cont.)
 Example: Identify the graph and parts of
    
2 2
4 9 16 90 205 0
x y x y

12. Identifying Conic Sections (cont.)
 Example: Identify the graph and parts of
To identify the graph, we have to rewrite the equation:
    
2 2
4 9 16 90 205 0
x y x y
   
    
2 2
4 16 9 90 205
x x y y

13. Identifying Conic Sections (cont.)
 Example: Identify the graph and parts of
To sketch the graph, we have to rewrite the equation:
    
2 2
4 9 16 90 205 0
x y x y
   
    
2 2
4 16 9 90 205
x x y y
       
    
 
 
2 2
2
2
2 2
4 4 10 205
2 4 2
9 5 9 5
x x y y
Remember, you have to
factor out the
coefficients of x2 and y2!

14. Identifying Conic Sections (cont.)
 Example: Identify the graph and parts of
To identify the graph, we have to rewrite the equation:
    
2 2
4 9 16 90 205 0
x y x y
   
    
2 2
4 16 9 90 205
x x y y
       
    
 
 
2 2
2
2
2 2
4 4 10 205
2 4 2
9 5 9 5
x x y y
   
   
2 2
4 2 9 5 36
x y

15. Identifying Conic Sections (cont.)
 Example: Identify the graph and parts of
To identify the graph, we have to rewrite the equation:
    
2 2
4 9 16 90 205 0
x y x y
   
    
2 2
4 16 9 90 205
x x y y
       
    
 
 
2 2
2
2
2 2
4 4 10 205
2 4 2
9 5 9 5
x x y y
   
 
 
2 2
4 2 9 5 36
36 36 36
x y

16. Identifying Conic Sections (cont.)
 Example: Identify the graph and parts of
To identify the graph, we have to rewrite the equation:
    
2 2
4 9 16 90 205 0
x y x y
   
    
2 2
4 16 9 90 205
x x y y
       
    
 
 
2 2
2
2
2 2
4 4 10 205
2 4 2
9 5 9 5
x x y y
   
 
 
2 2
4 2 9 5 36
36 36 36
x y
 
   
 
   
   
2 2
2 5
1
3 2
x y
   
 
 
2 2
2 5
1
9 4
x y


17. Identifying Conic Sections (cont.)
 Example (cont.):
This is an ellipse.
The center is at 2, –5
The x-radius is 3 (semi-major)
The y-radius is 2 (semi-minor)
 
   
 
   
   
2 2
2 5
1
3 2
x y

18. Identifying Conic Sections (cont.)
 Example: Identify     
2 2
9 4 90 32 197 0
x y x y

19. Identifying Conic Sections (cont.)
 Example: Identify     
2 2
9 4 90 32 197 0
x y x y
       

  
 

 
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x x y y
Notice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

20. Identifying Conic Sections (cont.)
 Example: Identify     
2 2
9 4 90 32 197 0
x y x y
       

  
 

 
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x x y y
   
    
2 2
9 6
4 4
5 3
x y
Notice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

21. Identifying Conic Sections (cont.)
 Example: Identify     
2 2
9 4 90 32 197 0
x y x y
       

  
 

 
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x x y y
   
  
 
  
2 2
9 5 4 4 36
36 36 36
x y
Notice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

22. Identifying Conic Sections (cont.)
 Example: Identify     
2 2
9 4 90 32 197 0
x y x y
       

  
 

 
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x x y y
   
  
 
  
2 2
9 5 4 4 36
36 36 36
x y
   
 
  
2 2
5 4
1
4 9
x y
Notice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

23. Identifying Conic Sections (cont.)
 Example: Identify
 The squared terms have opposite signs, so this is a
hyperbola.
    
2 2
9 4 90 32 197 0
x y x y
       

  
 

 
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x x y y
   
  
 
  
2 2
9 5 4 4 36
36 36 36
x y
   
 
  
2 2
5 4
1
4 9
x y
   
 
  
2 2
2 2
5 4
1
2 3
x y
Notice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

24. Identifying Conic Sections (cont.)
 Example: Identify
Center –5, 4
    
2 2
9 4 90 32 197 0
x y x y
   
 
  
2 2
2 2
5 4
1
2 3
x y

25. Identifying Conic Sections (cont.)
 Example: Identify
Center –5, 4
opens in y-direction
rx = 2, ry = 3
 vertices 3
    
2 2
9 4 90 32 197 0
x y x y
   
 
  
2 2
2 2
5 4
1
2 3
x y

26. Identifying Conic Sections (cont.)
 Example: Identify
Center –5, 4
opens in y-direction
rx = 2, ry = 3
 vertices 3
slope of asymptotes:
    
2 2
9 4 90 32 197 0
x y x y
   
 
  
2 2
2 2
5 4
1
2 3
x y

3
2

27. Identifying Conic Sections (cont.)
 Example: Identify
Center –5, 4
opens in y-direction
rx = 2, ry = 3
 vertices 3
slope of asymptotes:
    
2 2
9 4 90 32 197 0
x y x y
   
 
  
2 2
2 2
5 4
1
2 3
x y

3
2

28. Classwork
 10.4 Assignment (College Algebra)
 Page 985: 2-22 (even); page 978: 28-36 (even);
page 969: 44-48 (even)
 10.4 Classwork Check
 Quiz 10.3