Section 6-6:  Systems of Second-Degree Equations

Try the quiz at the bottom of the page!
go to quiz
 
 
There are many methods for solving a system of second-degree equations in two variables.  In this section we will concentrate on the algebraic approach using substitution and/or elimination.  We have talked about solving them using a graphing caluculator.

 
1)  Solve the system x2 + y2 = 20 and (x - 5)2 + (y - 5)2 = 10.
 
                 Solution:  Write both in expanded form:
x2 + y2 = 20
x2 - 10x + y2 - 10y = -40
Subtract the two equations to get:
-10x - 10y = -60
Divide by -10
x + y = 6
This line represents the line containing any intersection points of the two circles.  Isolate for either x or y.
y = 6 - x
now substitute back into one of the original equations.  Use the top one
x2 + (6 - x)2 = 20
x2 + 36 - 12x + x2 = 20
2x2 - 12x + 16 = 0
x2 - 6x + 8 = 0
(x - 4)(x - 2) = 0
x = 4 or x = 2
To find y, use the red equation above.
When x = 4, y = 2
When x = 2, y = 4
The intersection points are (4, 2) and (2, 4)
 
2)  Find the intersection of 3x2 + y2 = 15 and x2 - y2 = 1.
 
                 Solution:  The first equation is an ellipse and the second is a hyperbola.  Add the two equations to get:
4x2 = 16
x2 = 4
x = 2 or x = -2
Now replace these answers in one of the above equations to find the y values.
4 - y2 = 1
-y2 = -3
y2 = 3
y = 1.7 or y = -1.7
The intersection points are (2, 1.7), (2, -1.7), (-2, 1.7), (-2, -1.7)
 
3)  Find the intersection of  x2 + y2 = 1 and x2 + 4y2 = 13.
 
                 Solution:  Subtract these two equations to get:
3y2 = 12
y2 = 4
y = 2 or y = -2
Now put these into one of the original equations.  Use the first one.
x2 + 4 = 1
x2 = -3
This means that the answers are imaginary.  What does that mean about the intersection?  You are right!  No intersection.  Here is the graph:

 
 
Bring on the sample test: 
 
Let me restudy: 
 

 
 
Current quizaroo #  6
 
1) Find the center point and radius for the circle:  x2 + 4x + y2 - 6y - 23 = 0  
 
a)  (-2, 3), with radius 36
b)  (-2, 3) with radius 6
c)  (2, -3) with radius 36
d)  (2, -3) with radius 6
e)  (2, 3) with radius 6
 
 
 
2)  Which of the following is a vertex point for the ellipse  4(x - 1)2 + 25(y - 2)2 = 100

          a)  (3, 2) 

b)  (1, 4)
c)  (1, 7)
d)  (6, 2)
e)  (6, 4)
 

 
 
3)  Which one is an equation of an asymptote for the hyperbola:  (x - 1)2 - (y - 3)2 = 36
 
a)  y - 3 = -1(x - 1)
b)  y = x
c)  y = -x
d)  y - 3 = -6(x - 1)
e)  y - 3 = 6(x - 1)
 
 
 
4)  A parabola is the set of all points equdistant from a fixed point to a fixed line.  The fixed line
 is called?
a)  latus rectum
b)  chord
c)  directrix
d)  focus
e)  major axis
 
 
  5)  What is the most number of times a hyperbola can intersect a circle?
 
a) 2
b) 3
c) 4
d) 5
e) infinite number
 
 
   
 click here for answers!!