Name: _________________________________
Pre-Calculus Guided Notes: Chapter 10 – Conics
Section 2 – Circles
A circle is _______________________________________________________________________________
__________________________________________________________________________________________
Example 1
Write an equation for the circle with center (3, 2) and radius 5. To do this, we’ll need the
distance formula: d 
x2  x1 2   y2  y1 2
In general:
An equation of the circle with center (h, k) and radius r (r > 0) is __________________________.
If the center is the origin, then ________________________. The general form of the equation
of a circle is x2 + y2 + Dx + Ey + F = 0, where D, E, and F are constants.
Example 2
Sketch the graph of x2 + y2 – 8x + 2y + 8 = 0.
Center _______
Radius _______
1
Example 3
Sketch the graph of x2 + y2 6x – 2y – 6 = 0.
Center ___________
Radius ___________
Example 4
Sketch the graph of 2x2 + 2y2 = 18.
Center ___________
Radius ___________
Example 5
Sketch the graph of x2 + y2 + 6y = 0.
Center ___________
Radius ___________
Example 6
Write the standard form of the equation of the circle that is tangent to the x-axis and has
its center at (-5, 4).
2
Section 3 – Ellipses
An ellipse is _____________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Each of the two fixed points is a _______________ (plural: __________) of the ellipse and the
distances from the foci to a point P on the curve are called the ___________________________.
Example 1
Find the equation for an ellipse with foci F1(-4, 0) and F2(4, 0) and PF1 +PF2 = 10.
3
Foci on the x-axis (c, 0) and (-c, 0)
and center at the origin
Equation:
Foci on the y-axis (0, c) and (0, -c)
and center at the origin
Equation:
The sum of the focal radii for each of its points is the constant 2a (a > c)
c2 = a2 – b2
x-intercepts:
x-intercepts:
y-intercepts:




y-intercepts:
In each case, the ellipse is symmetric with respect to both the x-axis and the y-axis and a > b
Major axis: the segment of length 2a cut off by the ellipse
Minor axis: the segment of length 2b that is perpendicular to the major axis at the center
Vertices: the points where the ellipse cuts it major axis
Example 2
Sketch the graph of 4x2 + 25y2 = 100.
horizontal/vertical ______
shift (h, k)
______
vertices
______
foci
______
major axis length
______
minor axis length
______
An ellipse can have its center at a point other than the origin, and its axis need not lie on
the coordinate axes.
Graphs of the following equations have center (h, k) and a > b.
The foci are located c units to either side of the center along the major axis.
x  h2   y  k 2  1
x  h2   y  k 2  1
b2
a2
b2
a2
4
Example 3
Sketch the graph of
x  32   y  42
64
100
 1.
horizontal/vertical ______
shift (h, k)
______
vertices
______
foci
______
major axis length
______
minor axis length
______
Example 4
Sketch the graph of the ellipse with equation 4x2 + 9y2 – 8x – 54y + 49 = 0.
horizontal/vertical ______
shift (h, k)
______
vertices
______
foci
______
major axis length
______
minor axis length
______
Example 5
Consider the graphed ellipse. Write the equation of the ellipse in standard form and find
the coordinates of the foci.
5
Example 6
Determine an equation for an ellipse on the coordinate axes with major axis of length 10
and foci at (3, 0), and (-3, 0).
Section 4 – Hyperbolas
A hyperbola is ___________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
Each fixed point is called a ________________ and the distances from the foci to a point P on
the curve are called ________________________.
Example 1
Write an equation for the hyperbola with foci F1(-5, 0) and F2(5, 0) and with focal radii
differing by 8.
6
Graphs of the following equations have center (h, k).
The foci are located c units to either side of the center along the transverse axis.
c2 = a2 + b2
2
2
x  h    y  k   1
 y  k 2  x  h2  1
a2
b2
a2
b2
Example 2
Sketch the graph of 9x2 – 25y2 = 225.
horizontal/vertical ______
Example 3
Sketch the graph of
______
vertices
______
foci
______
transverse axis length
______
conjugate axis length
______
asymptote equations
______
 y  12  x  52
9
shift (h, k)
25
 1.
horizontal/vertical ______
shift (h, k)
______
vertices
______
foci
______
transverse axis length
______
conjugate axis length
______
asymptote equations
______
7
Example 4
Graph 4x2 – y2 + 24x + 4y + 28 = 0.
horizontal/vertical ______
shift (h, k)
______
vertices
______
foci
______
transverse axis length
______
conjugate axis length
______
asymptote equations
______
Example 5
Find the equation of the hyperbola with foci at (1, -5) and (1, 1) and whose transverse axis
is 4 units long.
Rectangular Hyperbola: xy = c
Example 6
Graph xy = 36.
8
Section 5 – Parabolas
A parabola is ____________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
The fixed line is called the __________________ and the fixed point is called the _____________.
Example 1
Write an equation for the parabola with focus F(0, 4) and directrix the line L with equation
y = -2.
Example 2
Write and equation for the parabola with focus F(3, 2) and directrix the line L with equation
x = -1.
9
Notice that the vertex of a parabola is ____________________________________________________
__________________________________________________________________________________________
Parabola w/ vertex (h, k) and directrix y = k – p
(p is the distance between the focus and the vertex)
x
Parabola w/ vertex (h, k) and directrix x = h – p
(p is the distance between the focus and the vertex)
1
 y  k 2  h
4p
y
x = a(y – k)2 + h
1
x  h2  k
4p
y = a(x – h)2 + k
Example 3
Graph x + 1 = 4y – y2
horizontal/vertical ______
P
______
vertex (h, k)
______
focus
______
directrix equation
______
Example 4
Graph x2 – 8x – y + 18 = 0
horizontal/vertical ______
P
______
vertex (h, k)
______
focus
______
directrix equation
______
10
Example 5
Write an equation for the parabola with a focus at (-1, 7), the length from the focus to the
vertex is 2 units, and has a minimum.
Section 6 – Rectangular and Parametric Forms of Conic Sections
The equation of a conic section can be written in the form:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0,
where A, B, and C are not all zero.
In general, the graph of
Ax2 + Cy2 + Dx + Ey + F = 0
is a:
when…
Circle
Parabola
Ellipse
Hyperbola
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
1. 4x2 – 9x + y – 5 = 0
2. 4x2 – y2 + 8x – 6y + 4 = 0
3. 2x2 + 4y2 – 4x + 12y = 0
4. 2x2 + 2y2 – 8x + 12y + 2 = 0
11
So far we have discussed equations of conic sections in their rectangular form. Some conic
sections can also be described parametrically.
Example 1
 x  t  1
Graph the curve defined by the parametric equations 
, where  2  t  2 . Then
 y   4  t 2
identify the curve by finding the corresponding rectangular equations.
t
x
y
-2
-1
0
1
2
Example 2
 x   cos t
Find the rectangular equation of the curve whose parametric equations are 
2
 y  sin t
where 0 o  t  180 o . Then graph the equation using arrows to indicate how the graph is
traced.
Example 3
x2 y2
Find parametric equations for the equation

 1.
9
4
12
Section 7 – Transformations of Conics
Remember from earlier in this course that Th,k refers to a translation of h units horizontally
and k units vertically.
Example 1
Given 2x2 + 3xy – 4y2 – 5x = 0. Write the equation following a translation of T3, 1 in general
form.
Another type of transformation we studied this year is rotations. The figures below show
an ellipse whose center is the origin and its rotation.
A rotation of  about the origin can be described by the matrix:
_____________________
If we let P(x, y) be a point on the graph of a conic section, then P’(x’, y’) is the image of P
after a counterclockwise rotation of  . The values of x’ and y’ can be found by matrix
multiplication:
13
Rotation Equations
To find the equation of a conic section with respect to a rotation of  , replace
x with ________________________ and y with ____________________________
Example 2
Given x2 + 4xy + 5y2 – 7x – y = 0. Write the equation of the graph after a rotation of  =90o.
Identifying Conics by Using the Discriminant
For the general equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0,

if B2 – 4AC < 0, the graph is a circle (A = C, B = 0) or an ellipse (A  C or B  0).

if B2 – 4AC > 0, the graph is a hyperbola.

if B2 – 4AC = 0, the graph is a parabola.
Example 3
Identify the graph of the equation 8x2 + 5xy – 4y2 = -2.
14
Angle of Rotation About the Origin
For the general equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the angle of rotation  about
the origin can be found by


4
if A = C, or
tan 2 
B
, if A  C
AC
Example 4
Identify the graph of the equation x2 – 4xy – 2y2 – 6 = 0. Then find  .
15