Chapter 8 Conic Sections
8.1 Distance and Midpoint
A.
OBJ: Find the length and midpoint of a line segment
Facts/Formulas:
2
2
The distance formula d  ( x 2  x1 )  ( y 2  y1 ) gives the distance d between the points (x1, y1)
B.
and (x2, y2).
 x1  x 2 y1  y 2 
,

2
2  gives the midpoint M of the line segment joining
The midpoint formula M 
A(x1, y1) and B(x2, y2).
C. Examples:
1.Find the distance between (2, 4) and (3, 4).
2.Classify ΔABC as scalene, isosceles, or equilateral. A(1,5), B(3,1), C(9,4)
3. Find the midpoint of the line segment joining (3, 1) and (2, 5).
4. The points O(0, 0), A(4, 4), and B(4, 4) lie on a circle. Find the diameter of the circle.
8.2 Graph and Write Equations of Parabolas
A. OBJ: To graph and write equations of parabolas that open up/down and left/right.
B. Facts/Formulas:
a. On any parabola, each point is equidistant from a point called the focus and a line
called the directrix (perpendicular to axis of symmetry).
b. Vertex – ½ way between focus and directrix.
c. Parabola that opens UP or DOWN: y=ax2 or x2 = 4py
i. if a > 0 (is +), then the parabola opens up.
ii. If p > 0 (is +), then the parabola opens up.
d. Parabola that opens Right or Left: y2= 4px
i. If p > 0 (is +), then the parabola opens to the right.
e. Standard Equation with Vertex @ (0, 0)
Equation
Focus
Directrix
Axis of
Symmetry
2
x = 4py
(0, p)
Y = -p
Vertical (x=0)
2
y = 4px
(p, 0)
X = -p
Horizontal (y=0)
f.
C. Examples:
8.3 Graph and Write Equations of a Circle
A. OBJ: to graph and write the equation of a circle.
B.Facts/Formulas:
1. A circle is the set of all points (x, y) in a plane that are equidistant from a fixed point,
called the center of the circle. The distance r between the center and any point (x, y) on
the circle is the radius.
2. Equation of a Circle: x2 + y2 = r2 with center @ (0, 0)
C.Examples:
8.4 Graph and Write Equation of an Ellipse
A. OBJ: to graph and write an equation of an ellipse.
B. Facts/ Formulas:
1. An ellipse is the set of all points P in a plane such that the sum of the distances between P
and two fixed points, called the foci, is a constant.
2. The line through the foci intersects the ellipse at two vertices. The major axis joins the
vertices. Its midpoint is the ellipse's center.
3. The line perpendicular to the major axis at the center intersects the ellipse at the two covertices, which are joined by the minor axis.
4.
C.Ex:
8.5 Graph and Write equations of Hyperbolas
A.OBJ: to graph and write equations of hyperbolas
B.Facts/Formulas:
1. A hyperbola is the set of all points P in a plane such that the difference of the
distances between P and two fixed points, again called the foci, is a constant.
2. The line through the foci intersects the hyperbola at the two vertices. The transverse
axis joins the vertices. Its midpoint is the hyperbola's center.
3. P. 518
C.Ex:
8.6 Translations with Conics
A. OBJ: to shift the center of a conic vertically and/or horizontally.
B. Facts/Formulas:
1.