Introduction to Conic Sections

Show your work for each problem.

1.
What type of conic section is ?
1. a)Parabola
2. b)Ellipse
3. c)Hyperbola
4. d)Circle
Answer: Circle
This is a circle because both and have equal coefficients (both 1) and are added together. The radius is .
2.
What type of conic section is ?
1. a)Hyperbola
2. b)Parabola
3. c)Circle
4. d)Ellipse
Answer: Parabola
This is a parabola because only is squared while is linear. Parabolas always have exactly one squared variable.
3.
What is the radius of the circle ?
Answer: 6
Since , we have . The radius is 6 units.
4.
What type of conic section is ?
1. a)Circle
2. b)Ellipse
3. c)Parabola
4. d)Hyperbola
Answer: Ellipse
This is an ellipse because both terms are positive (addition) and have different denominators (). If they were equal, it would be a circle.
5.
What type of conic section is ?
1. a)Parabola
2. b)Circle
3. c)Ellipse
4. d)Hyperbola
Answer: Hyperbola
This is a hyperbola because there is subtraction between the squared terms. The minus sign is the key identifier for hyperbolas.
6.
For the ellipse , what is the value of the semi-major axis ?
Answer: 7
The semi-major axis is the square root of the larger denominator. Since , we have , so .
7.
Identify the conic section and find its key features:
Answer: circle, radius 7
1. Are the coefficients of and equal? yes
2. Is there addition or subtraction between the terms? addition
3. What type of conic is this? circle
4. What is the radius? (r = sqrt of 49) 7
8.
Which direction does the parabola open?
1. a)Up
2. b)Left
3. c)Down
4. d)Right
Answer: Left
Since is squared (not ), the parabola opens horizontally. The negative coefficient means it opens to the left.
9.
Identify the conic and determine its orientation:
Answer: vertical hyperbola
1. Is there addition or subtraction between the terms? subtraction
2. What type of conic has subtraction between squared terms? hyperbola
3. Which term is positive (comes first)? y^2
4. Since is positive, the hyperbola opens... up and down
10.
For the ellipse , what is the length of the major axis?
Answer: 20
Since , we have , so . The major axis length is .
11.
How many types of conic sections are there?
1. a)4
2. b)3
3. c)5
4. d)2
Answer: 4
There are 4 conic sections: circle, ellipse, parabola, and hyperbola. Each is formed by slicing a cone at a different angle.
12.
Analyze the equation and identify the conic with its radius.
Answer: circle, radius 4
1. Divide both sides by 4 to simplify. What do you get? x^2 + y^2 = 16
2. Are the coefficients of and now equal? yes
3. What type of conic is this? circle
4. What is the radius? () 4
13.
A parabola has the equation . If the focus is at where , what is ?
Answer: 3
Comparing , we get , so . The focus is at .
14.
Classify and analyze:
Answer: ellipse, a=3, b=2
1. Divide by 36 to get standard form. What is the term? x^2/4
2. What is the term? y^2/9
3. Is this a circle or ellipse? ellipse
4. What is (from larger denominator 9)? 3
15.
Which conic section represents the path of a planet orbiting the Sun?
1. a)Circle
2. b)Hyperbola
3. c)Ellipse
4. d)Parabola
Answer: Ellipse
According to Kepler's First Law, planets orbit the Sun in elliptical paths with the Sun at one focus. A circle is a special case of an ellipse.
16.
How many squared variables does a parabola equation have?
Answer: 1
A parabola has exactly 1 squared variable. For example, has only squared, while has only squared.
17.
Determine if is a circle or ellipse, and find the radius if it's a circle.
Answer: circle, radius 2
1. Divide both sides by 25. What do you get? x^2 + y^2 = 4
2. Are the coefficients of and equal? yes
3. So this is a...? circle
4. What is the radius? (sqrt of 4) 2