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?
