Precalculus
Notes: Systems of Conic Sections
Name
Date
A system of equations is a set of two or more equations.
You can solve a system algebraically (using substitution or elimination) or graphically (by
sketching the graphs on a coordinate plane).
The solution(s) to a system are the points where the graphs of the equations intersect.
It is possible for the system to have no solution if the graphs never intersect one another.
Review: Solve the system of linear equations using the 3 methods (substitution, elimination, and graphically).
2x − y = 8
−3 x + y = −7
Substitution
Elimination
Graphically
Precalculus
Notes: Systems of Conic Sections
Name
Date
Solve the system algebraically using the substitution method. If there are no solutions, write no solution.
Ex. 1
x2 + y 2 = 9
16 x 2 + 9 y 2 = 144
Ex. 2
y = x2 + 8x + 9
y = x+3
Solve the system algebraically using the elimination method. If there are no solutions, write no solution.
Ex. 3
9 x 2 + 4 y 2 = 36
2 x2 − 2 y 2 = 8
Ex. 4
−2 y 2 + x 2 = 17
y 2 + 2 x 2 = 54
Solve the system graphically. If there are no solutions, write no solution.
Ex. 5
x 2 + y 2 = 25
x2 + y2 = 5
x + 2y = 5
Ex. 6
2 x2 − y 2 = 2
6
4
4
2
2
-5
-5
5
5
-2
-2
-4
-4
-6
y = − x2 + 3
Ex. 7
5 x 2 + 5 y 2 = 125
y = x2 + 1
3 x 2 + 3 y 2 = 27
Ex. 8.
6
6
4
4
2
2
-5
5
-2
-4
-2
-6
Precalculus
Chapter 10 Worksheet #8 Systems of Conic Sections
Name
Date
Solve each system of equations algebraically. Check your answers.
If there are no solutions, write no solution.
1.
2 x 2 + 2 y 2 = 52
y = x−4
3.
x 2 − 4 y 2 = 16
1
y = x +1
2
5.
7.
x 2 − y 2 = 25
x 2 + y 2 = 25
x 2 + y 2 = 25
x − y =1
2.
4.
6.
8.
x 2 + y 2 = 10
x + y = −2
x2 − y = 0
y = x2 − 4x + 4
2 x 2 − 3 y 2 = −1
2x2 + 3 y2 = 5
x2 − y 2 = 9
x 2 + 9 y 2 = 169
Solve each system of equations graphically. Check your answers. If there are no solutions, write no solution.
x 2 + y 2 = 20
y − x = −2
9.
10.
2 y = x2
y+x=4
6
8
4
6
2
4
-5
5
2
-2
-5
5
-4
-2
11.
5 y2 = 5
12.
3x 2 + 3 y 2 − 3 = 0
2
5
-2
10
2 x 2 − 3 y 2 = −1
2x2 + 3 y2 = 5
Precalculus Chapter 10
Worksheet Review A Systems of Conic Sections
Name
Date
Solve each system of equations algebraically or graphically. Check your answers.
If there are no solutions, write no solution.
1.
3.
4 x2 − 2 y2 = 4
x 2 + y 2 = 25
x2 + y 2 = 9
4 x 2 − y 2 = 16
2.
4.
16 y 2 − 9 x 2 − 128 y + 18 x + 103 = 0
8 y 2 + 18 x 2 − 64 y − 36 x + 74 = 0
y = 5− x
y = x2 − 6 x + 9
8
6
4
2
5
-2
10
Precalculus Chapter 10
Worksheet Review B Systems of Conic Sections
Name
Date
Solve each system of equations algebraically or graphically. Check your answers.
If there are no solutions, write no solution.
1.
3.
x2 + y 2 = 4 x − 3
x2 − y2 = 1
x2 + y 2 = 9
−2 x 2 + 2 y 2 = −6
2.
4.
x 2 + 64 y 2 = 64
x 2 + y 2 = 64
x 2 + y 2 = 16
y = 2 x + 10
6
4
2
-5
5
-2
-4
-6
Precalculus
Warm Up: Systems of Conic Sections
1.
Name
Date
A line intersects a parabola. What is the minimum number of solutions to the system? Explain.
2. Is it possible for a parabola to intersect a circle at only one point? Explain.
3. Is it possible for a hyperbola to intersect an ellipse at only one point? Explain.
4. Suppose a hyperbola intersects an ellipse. If both conic sections are centered about the origin, what is the
minimum number of solutions to the system? Draw a picture to support your answer.
Precalculus
Warm Up: Systems of Conic Sections
1.
Name
Date
A line intersects a parabola. What is the minimum number of solutions to the system? Explain.
2. Is it possible for a parabola to intersect a circle at only one point? Explain.
3. Is it possible for a hyperbola to intersect an ellipse at only one point? Explain.
4. Suppose a hyperbola intersects an ellipse. If both conic sections are centered about the origin, what is the
minimum number of solutions to the system? Draw a picture to support your answer.
Precalculus
Exit Slip: Systems of Conic Sections
Name
Date
Solve each system of equations graphically. Check your answers.
If there are no solutions, write no solution.
1.
( x − 2)2
+ y2 = 1
9
x2 − 3 y2 = 1
y=
2.
1
x −3
2
( x + 2)
2
1
Precalculus
Exit Slip: Systems of Conic Sections
( y + 1)
+
2
=1
4
Name
Date
Solve each system of equations graphically. Check your answers.
If there are no solutions, write no solution.
1.
( x − 2)2
+ y2 = 1
9
2
x − 3 y2 = 1
y=
2.
1
x −3
2
( x + 2)
1
2
( y + 1)
+
4
2
=1