Name: _____________________________________
Secondary 2 Honors – Unit 12 Guided Notes
Date: ____________________
Period: ___________
12.1 Conic Sections: Parabolas
Conic Sections are figures that can be formed by slicing a three dimensional right circular cone with a plane. These
figures can be represented on the graph as well as algebraically. The four conic sections are circles, ellipses, parabolas,
and hyperbolas.
A parabola is the set of points that are equally distant from a focus point and the directrix, a fixed line. The
standard equation depends on the axis of symmetry.
•
A vertical axis (the parabola opens up or down) has a focus at _____________________ and the
equation ___________________________________.
•
A horizontal axis (opens left or right) has a focus at_____________________ and the
equation___________________________________.
•
The vertex is always halfway between the ___________ & ______________ at a distance 𝒑 from both.
Some things to note:
• __________________ is a point p units inside the parabola
•
__________________ is a line p units outside the parabola
• _____________________________ is the distance across the
parabola at the focus (4p).
• The distance from a point on the parabola to the ____________
is the same distance from that point to the focus.
Some helpful tips for finding equations of parabolas
1. Determine which form to use for the parabola by finding out which way the graph opens
•
________________________ if the directrix is in the form of 𝑦 =
•
________________________ if the directrix is in the form of 𝑥 =
2. Find the _______ (distance from vertex to focus or from the vertex to the directrix)
•
If given the focus and directrix, remember that p will be ½ of that distance between those two
3. Find the vertex – if not given
•
The vertex always has the same ____-coordinate as the focus if the parabola opens up or down
•
•
The y-coordinate will be the midpoint of the y-coordinates of the directrix and focus
The vertex always has the same ____-coordinate as the focus if the parabola opens left or right
•
The x-coordinate will be the midpoint of the x-coordinates of the directrix and focus
4. Find the focal diameter – if asked
• The focal diameter is always ______
Example 1: Find the equation of a parabola whose focus is at (−5,0) and whose directrix is 𝑥 = 5.
Example 2: Find the equation of a parabola whose vertex is at (2, −3) and whose focus is at (5, − 3).
List the vertex, focus, directrix and focal diameter and then draw the graph for each parabola:
3. 𝑥 2 = −12𝑦
4. (𝑥 − 1)2 = 8(𝑦 + 1)
5. 𝑦 2 − 6𝑦 − 12𝑥 + 33 = 0
__________________________________________________________________________________________
12.2 Conic Sections: Ellipses
An ellipse is formed by cutting a three dimensional cone with a slanted plane. This differs from a circle in that an ellipse
does not have a constant radius. It has a radius that changes in between an x radius and y radius. However, an ellipse
has two focal points in which the sum of the length of both focal points to any given point on the ellipse is always the
same.
In an ellipse there are 2 lengths of radii. The longer radius is called the
______________________________, whereas the shorter radius is called
the ________________________________.
Each ellipse has 7 major points:
 Center is given as __________________ – formula is given below
 Vertices (2) are the points on the ellipse on the _________________ axis
 Vertices are _________ distance from the center on major axis
 Co-vertices (2) are the points on the ellipse on the _______________ axis
 Co-vertices are _______ distance from the center on minor axis
 Foci (2) are _________ distance from center on the major axis.
 𝑐 is found from using the formula _________________________________
Equation of an ellipse with major axis horizontal:
Equation of an ellipse with major axis vertical:
Some things to note:
•
𝒂 is always ________________ than 𝒃
•
𝒂 is the ___________________ of the radius of the major axis and 𝒃 is the ____________________ of
the radius of the minor axis
•
________________________ is the measure of how round the ellipse is. This is found by ___________.
Some helpful tips for finding parts of ellipses
1. Determine which form to use for the ellipse by finding out which way the major axis is
•
If 𝒂 is below the (𝑥 − ℎ)2 , then the major axis is __________________________
•
If 𝒂 is below (𝑦 − 𝑘)2 then the major axis is ____________________________
•
If no equation is given look at the vertices or foci,
•
If the vertices or foci have the same _________________, then the major axis is horizontal
•
If the vertices of foci have the same __________________, then the major axis is vertical.
2. Find the center (ℎ, 𝑘) – if not given
•
This will always be the ________________________ of the vertices or foci
3. Find the c (distance from center to foci)
•
If foci is not given: use _______________________________________
4. Find 𝒂 and 𝒃
•
Find a by finding the ________________________ from the center to one of the vertices
•
Find b by using 𝑎2 = 𝑏 2 + 𝑐 2
Example 1: Find the equation of an ellipse with foci at (−3,0) 𝑎𝑛𝑑 (3,0) and vertices at (−4,0) 𝑎𝑛𝑑 (4,0).
Example 2: Find the equation of an ellipse with vertices (−7,3) 𝑎𝑛𝑑 (3,3) and foci at (−6,3) 𝑎𝑛𝑑 (2,3).
List the center, vertices, foci, and eccentricity and then draw the graph for each ellipse:
3.
𝑥2
9
+
𝑦2
36
=1
List the center, vertices, foci, and eccentricity and then draw the graph for each ellipse:
4.
5.
(𝑥−2)2
9
+
(𝑦+1)2
16
=1
4𝑥 2 + 24𝑥 + 25𝑦 2 − 50𝑦 − 39 = 0