Pre-Calculus Notes Chapter 8 Analytical Geometry in Two and Three Dimensions
8.1 Conic Sections and Parabolas
A parabola is the set of all points in a plane equidistance from a particular line (the directrix) and a
particular point (the focus) in the plane.
Parabolas with Vertex (0,0)
Standard Equation
Opens
Focus
Directrix
Axis
Focal Length
Focal Width
x² = 4py
Upward or Downward
(0,p)
y = -p
y-axis
p
|4p|
y²= 4px
To the left or to the right
(p,0)
x = -p
x-axis
p
|4p|
Find the focus, directrix, and the focal width of the parabola: y = -(1/2)x².
Parabolas with Vertex (h,k)
Standard Equation
Opens
Focus
Directrix
Axis
Focal Length
Focal Width
(x-h)² = 4p(y-k)
Upward or Downward
(h, k + p)
y =k -p
x =h
p
|4p|
(y-k)²= 4p(x-h)
To the left or to the right
(h + p,k)
x =h -p
y=k
p
|4p|
Find the standard form of the equation for the parabola with vertex (3,4) and focus (5,4) .
Prove that the graph of y² - 6x + 2y + 13 = 0 is a parabola and find its vertex, focus and directrix.
Assignment: Page 641 # 5 – 40 (5s)
8.2 Ellipses
An ellipse is the set of all points in a plane whose distances from two fixed points in the plane have a
constant sum. The fixed points are the foci (plural of focus) of the ellipse. The line through the foci is the
focal axis. The point on the focal axis midway between the foci is the center. The points where the
ellipse intersects its axis are the vertices of the ellipse.
Ellipses with Center (0,0)
Standard Equation
Focal Axis
Foci
Vertices
Semi major Axis
Semi minor Axis
Pythagorean Relation
x-axis
(
)
y-axis
(0, c)
a
b
a² = b² + c²
a
b
a² = b²+ c²
Find the vertices and the foci of the ellipse 4x² + 9y² = 36.
Ellipses with Center (h,k)
Standard Equation
Focal Axis
Foci
Vertices
Semi major Axis
Semi minor Axis
Pythagorean Relation
y=k
(h , k)
(h a, k)
a
b
a² = b² + c²
x =h
(h, k c)
(h, k a)
a
b
a² = b² + c²
Find the standard form of the equation for the ellipse whose major axis has endpoints ( -2,-1) and (8, -1)
and whose minor axis has length 8.
The eccentricity of an ellipse is
e=
√
where a is the semi major axis, b is the semi minor axis, and x is the distance from the center of the
ellipse to either focus.
Earth’s orbit has a semi major axis a
Calculate and interpret b and c.
Assignment: Page 653 # 5 – 40 (5s)
Page 641 # 45
Gm (gigameters) and an eccentricity of e 0.0167.
8.3 Hyperbolas
A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a
constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal
axis. The point on the focal axis midway between the foci is the center. The points where the hyperbola
intersects its focal axis are the vertices of the hyperbola.
How to Sketch the Hyperbola:
1. Sketch line segments at x = a and y = b, and complete the rectangle they determine.
2. Sketch the asymptotes by extending the rectangle’s diagonals.
3. Use the rectangle and asymptotes to guide your drawing.
Hyperbolas with Center (0,0)
Standard Equation
Focal Axis
Foci
Vertices
Semitransverse Axis
Semiconjugate Axis
Pythagorean Relation
Asymptotes
x-axis
( , 0)
( a, 0)
a
b
a² + b² = c²
y=
Find the vertices and the foci of the hyperbola 4x² - 9y²=36.
y-axis
(0, c)
(0, a)
a
b
a² + b² = c²
Hyperbolas with Center (h,k)
Standard Equation
Focal Axis
Foci
Vertices
Semitransverse Axis
Semiconjugate Axis
Pythagorean Relation
Asymptotes
y=k
(h , k)
(h a, k)
a
b
a² + b² = c²
x =h
(h, k c)
(h, k a)
a
b
a² + b² = c²
y=
Find the standard form of the equation for the hyperbola whose transverse axis has endpoints ( -2,-1)
and (8,-1) and whose conjugate axis has length 8.
The eccentricity of a hyperbola is
√
Where a is the semitranverse axis, b is the semiconjugate axis, and c is the distance from the center to
either focus.
A comet following a hyperbolic path about the sun has a perihelion distance of 90 Gm. When the line
from the comet to the Sun is perpendicular to the focal axis of the orbit, the comet is 281.25Gm from
the Sun. Calculate a, b, c, and e. What are the coordinates of the center of the Sun if we coordinatize
space so that the hyperbola is given by
?
Assignment: Page 663 # 5 – 40 (5s)
Page 654 # 45
Page 641 # 50
8.6 Three-Dimensional Cartesian Coordinate System
Important features of the three-dimensional Cartesian coordinate system:




The axes are labeled x, y, and z, and these three coordinate axes form a right-handed coordinate
frame: When you hold your right hand with fingers curving from the positive x-axis toward the
positive y-axis, your thumb points in the direction of the positive z-axis.
A point P in space uniquely corresponds to an ordered triple (x,y,z) of real numbers. The
numbers x, y, and z are the Cartesian coordinates of P.
Points on the axes have the form (x, 0, 0),( 0,y,0), and (0,0,z) with (x, 0,0) on the x -axis, (0,y,0) on
the y-axis and (0,0,z) on the z-axis.
The axes are paired to determine the coordinate planes:
o The coordinate planes are the xy-plane, the xz-plane, and the yz-plane, and have
equations z = 0, y = 0, and x = 0 respectively.
o Points on the coordinate planes have the form (x,y,0),(x,0,z), and (0,y,z) with the first
triple being on the xy-plane, the second triple on the xz-plane and the third on the yzplane.
o The coordinate planes meet at the origin (0,0,0).
o The coordinate planes divide space into eight regions called octants, with the first
octant containing all points in space with three positive coordinates.
The Distance Formula (Cartesian Space):
The distance f(P,Q) between the points P(
) and Q(
) in space is
d(P,Q) = √
The Midpoint Formula:
The midpoint M of the line segment PQ with endpoints P(
M=(
) and Q(
) is
)
Find the distance between the points P(-2,3,1) and Q(4,-1,5) and find the midpoint of the line segment
PQ.
The Standard Equation of a Sphere:
A point P(x,y,z) is on the sphere with center (h,k,l) and radius r if and only if
(x - h)²+(y - k)²+(z - l)² = r²
The standard equation of the sphere with center (2,0,-3)and radius 7 is:
Equation for a Plane in Cartesian Space:
Every plane can be written as:
Ax + By +Cz + D = 0,
Where A, B, and C are not all zero. Conversely, every first-degree equation in three variables represents
a plane in Cartesian space.
Sketch the graph 12x + 15y + 20z = 60.
Vector Relationships in Space:
For vectors v = <
> and w = <
>,
Equality: v = w if and only if
Addition: v + w = <
Subtraction: v – w = <
Magnitude: |v|=√
>
>
Dot Product: v w =
Unit Vector: u = v/|v|, v does not equal 0, is the unit vector in the direction of v.
Computing with vectors:
a) 3<-2,1,4>
b) <0,6,-7> + <-5,5,8>
c) <1,-2,4> - <-2,-4,5>
d) |<2,0,6>|
e) <5,3,-1> <-6,2,3>
Assignment: Page 693 # 5 – 40 (5s) SKIP 35
Page 664 # 45
Page 654 # 50
Page 641 # 55
Chapter 8 Review Assignment page 696 # 3-30 (3s), 37-47 (Odds), 63-73(Odds)