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 x^{2} + y^{2}
= 20 and (x - 5)^{2} + (y - 5)^{2} = 10.

Solution: Write both in expanded form:

x^{2} + y^{2 }= 20

x^{2} - 10x
+ y^{2} - 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

x^{2} + (6
- x)^{2} = 20

x^{2} + 36
- 12x + x^{2} = 20

2x^{2} -
12x + 16 = 0

x^{2} - 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 3x^{2} + y^{2} = 15 and x^{2} - y^{2}
= 1.

Solution: The first equation is an ellipse
and the second is a hyperbola. Add the two equations to get:

4x^{2 }=
16

x^{2} = 4

x = 2 or x = -2

Now replace these answers in one of the above equations to find
the y values.

4 - y^{2} = 1

-y^{2} =
-3

y^{2} = 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 x^{2} + y^{2} = 1 and x^{2} + 4y^{2} = 13.

Solution: Subtract these two equations
to get:

3y^{2} =
12

y^{2} = 4

y = 2 or y = -2

Now put these into one of the original equations. Use the
first one.

x^{2} + 4
= 1

x^{2} = -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: x^{2} + 4x + y^{2} - 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?