Advanced Algebra/Pre-calculus
Conic Sections
Name ______________________
Date ______________Block ____
1.2 “Circles”
Objectives:
1. To find the center, radius and equation of a circle given a graph or facts about the circle
2. To write the equation of a circle given a graph or facts about the circle
3. To find the center, radius and intercepts of a circle given its equation
Circle –The set of points in the xy-plane that are a fixed distance r from a fixed point (h, k). The fixed
distance r is called the radius, and the fixed point (h, k) is the center of the circle.
Standard form of the equation of a circle: ( x  h) 2  ( y  k ) 2  r 2 with center (h, k) and radius r.
General form of the equation of a circle:
x 2  y 2  ax  by  c  0
Examples: Write the standard form of the equation of the circle described.
y
1.
x
(4, 0)
**What do we notice about this circle??**
2.
Endpoints of a diameter: (1, 2) and (7, -6)
3. Change the answer from #2 to ‘general form’
Examples: a) Find the center and radius, b) graph, and c) find the intercepts, if any, of the graph
4.
( x  1) 2  ( y  4)2  9
y


x

















5. x 2  y 2  8 x 12 y  3  0
How do we find the center in this case? Remember from algebra 2 … complete the square!!!
Try to figure out what the last term will be in order to create perfect square trinomials:
Perfect Square Trinomial
x2 + 12x + _______
x2 – 14x + _______
x2 - 10x + _______
x2 + 5x + _______
=
Square of a Binomial
=
=
=
=
(
(
(
(
)2
)2
)2
)2
How did you determine what the last term would be to create the perfect square trinomial?
y

x 2  y 2  8 x 12 y  3  0













x























9.3 Ellipses Objectives:
1. To find all key components of an ellipse (center, foci, vertices, length of major and minor axes),
given the equation of an ellipse.
2. To sketch the graph of an ellipse, given the equation of the ellipse and vice versa.
Definition of an ellipse: An ellipse is the set of points in a plane, the sum of whose distances from two
fixed points, called foci, is a constant.
Important components of the ellipse:
 The line containing the foci is the major axis.
 The midpoint of the segment containing the foci is the center of the ellipse.
 The line through the center, perpendicular to the major axis, is the minor axis.
 The two points of intersection of the ellipse and the major axis are the vertices of the ellipse.
Equation of an ellipse with center at the origin:
Major axis
Foci
Vertices
x-axis
y-axis
(-c, 0) and (c, 0)
(0, -c) and (0, c)
(-a, 0) and (a, 0)
(0, -a) and (0, a)
**Easy ways to find Foci:**
a 2 + b 2 = c 2 or a 2 = c 2 - b 2 or b 2 = c 2 - a 2
Write your favorite equation of an ellipse….
Identify the important components….
Equation
x2 y 2

 1 Where a  b  0 and b2  a 2 - c 2
a 2 b2
x2 y 2
 2  1 Where a  b  0 and b2  a 2 - c 2
2
b
a
or even easier… Big2+Small2= Foci2
1. Discuss the ellipse (this means to find all components ~ center, vertices, foci and the lengths of the
major and minor axes. Sketch the graph.
9 x 2  16 y 2  144  0
Equation of an ellipse with center at (h, k) and major axis parallel to either x or y-axis:
Major axis Foci
Vertices
Equation
2
2
Parallel to
x  h  y  k 


 1 Where a  b and b2  a 2 - c 2
x-axis
(h + c, k ) and (h + a, k ) and (h - a, k )
2
2
a
b
(h - c, k )
Parallel to
y-axis
(h, k + c ) and
(h, k – c )
(h, k + a ) and (h, k - a)
 x  h
3. “Discuss” the ellipse and sketch the graph..
2( x  4)2  9( y  2)2  18
4. Determine the equation of the ellipse shown in the graph.
b
2
2
y k

a
2
2
 1 Where a  b and b2  a 2 - c 2
5. Determine the equation of the ellipse with foci at (0,  2) and length of major axis = 8.
6. Determine the equation of the ellipse with center (1, 2), vertex at (4, 2), containing the point (1, 3).
Assignment: P. 21 # 61 – 69 odd, 77 - 99 odd
Pp. 633-635, 13 – 16 all, 17 - 31 odd, 39-45 odd, 69, 73, 81