Identifying Conic Sections

Show your work for each problem.

1.
Identify the conic section:
1. a)Circle
2. b)Ellipse
3. c)Parabola
4. d)Hyperbola
Answer: Circle
This is a circle because both and have the same coefficient (1), and both terms are positive. The standard form is with .
2.
Identify the conic section:
1. a)Parabola
2. b)Hyperbola
3. c)Ellipse
4. d)Circle
Answer: Parabola
This is a parabola because only one variable () is squared. When only or only appears, it's always a parabola.
3.
Identify the conic section:
1. a)Circle
2. b)Hyperbola
3. c)Ellipse
4. d)Parabola
Answer: Ellipse
This is an ellipse because both terms are positive (added together), but the denominators are different (). If they were equal, it would be a circle.
4.
Identify the conic section:
1. a)Ellipse
2. b)Circle
3. c)Hyperbola
4. d)Parabola
Answer: Hyperbola
This is a hyperbola because the squared terms are subtracted (minus sign between them). Subtraction always indicates a hyperbola.
5.
Identify the conic section:
1. a)Parabola
2. b)Ellipse
3. c)Hyperbola
4. d)Circle
Answer: Hyperbola
This is a hyperbola because and are subtracted. Even though both coefficients are 1, the minus sign makes this a hyperbola, not a circle.
6.
For the equation , what is the coefficient A (for )?
Answer: 3
The coefficient A is 3 (the number multiplying ). Since , this equation represents a circle.
7.
For the equation , what is the coefficient C (for )?
Answer: -2
The coefficient C is -2. The minus sign in front of means . Since and have opposite signs, this is a hyperbola.
8.
In the equation , what is the value of when written in standard form ?
Answer: 9
Dividing by 36: simplifies to . So .
9.
What type of conic is ? Answer: circle, ellipse, parabola, or hyperbola
Answer: ellipse
This is an ellipse. Both and are positive (same sign), but (different values). Same sign + different values = ellipse.
10.
What type of conic is ? Answer: circle, ellipse, parabola, or hyperbola
Answer: parabola
This is a parabola. Only is squared (there's no term). This can be rewritten as , which is clearly a parabola.
11.
Identify the conic section:
Answer: circle
1. What is the coefficient A (for )? 1
2. What is the coefficient C (for )? 1
3. Since A = C = 1, what type of conic is this? circle
12.
Identify the conic section:
Answer: ellipse
1. What is the coefficient A (for )? 25
2. What is the coefficient C (for )? 4
3. Are A and C equal? no
4. Do A and C have the same sign? yes
5. Same sign but different values means what type of conic? ellipse
13.
Identify the conic section:
Answer: hyperbola
1. What is the coefficient A (for )? 4
2. What is the coefficient C (for )? Include the sign! -9
3. Do A and C have the same sign or opposite signs? opposite
4. Opposite signs means what type of conic? hyperbola
14.
Identify the conic section:
Answer: circle
1. What is the coefficient A (for )? 2
2. What is the coefficient C (for )? 2
3. Are A and C equal? yes
4. Equal positive coefficients means what type of conic? circle
15.
Identify the conic section:
Answer: parabola
1. Is there an term in this equation? no
2. Is there a term in this equation? yes
3. When only one variable is squared, what type of conic is it? parabola
16.
Identify the conic section:
Answer: ellipse
1. What is the coefficient A (for )? 9
2. What is the coefficient C (for )? 4
3. Are A and C both positive? yes
4. Are A and C equal? no
5. Same sign but different values means what conic? ellipse
17.
Identify the conic section:
Answer: hyperbola
1. What is the coefficient A (for )? 16
2. What is the coefficient C (for )? Include the sign! -25
3. Is A positive or negative? positive
4. Is C positive or negative? negative
5. Opposite signs means what type of conic? hyperbola
18.
Identify the conic section:
Answer: parabola
1. Is squared in this equation? no
2. Is squared in this equation? yes
3. How many variables are squared? 1
4. One squared variable means what type of conic? parabola