CONIC SECTIONS
After completing this topic, you should be able to:
1) identify the general shape of a conic section (circle, hyperbola, ellipse, parabola) produced by a
given equation
2) sketch the graph of a conic section if you are given an equation in standard form
3) Write the equation of a conic section given its graph
Before beginning the study of conic sections, review the general shape of the graph for functions that you
have already studied:
Name
linear
General form of the Equation
General shape of the graph
Ax + By = C
Ex: 4x + 5y = 20
quadratic
Ax2 + Bx + C = y
Ex: 3x2 + 2x - 8 = y
absolute value
A | Bx + C | = y
Ex: 2 | -3x + 5 | = y
logarithm
loga x = y
Ex: log3 6 = y
exponential
y = ax
Ex: y = 8x
conic section
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Ex: 2x2 + 4xy - y2 + 7y + 3 = 0 (note the
value D = 0 in this equation)
Now consider all equations of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 How does this general form
differ from the quadratic general form (Ax2 + Bx + C = y) ? (Look at the exponents of x and y).
_____________________________________________________________________________________
_____________________________________________________________________________________
1
You are correct if you noticed that both x and y are raised to the power of 2! To determine which conic
section (circle, ellipse or hyperbola) is produced, look at the examples below:
Equation
x2 + y2 = 8
4x2 + 4y2 = 10
2x2 + 2y2 = 4
x2 = 5 - y2
13x2 + 13y2 = 90
Graph
circle
circle
circle
circle
circle
What do all of these equations that produce a circle as a graph have in common (look the coefficient of the x2
and y2 terms when the equation is in general form)?
______________________________________________________________________________________
You are correct if you noticed that the coefficient of x2 and y2 is the same for each equation. So any equation
of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 in which A and C are the same will produce a circle when
graphed. The values B, D, E, and F will not affect the general shape (they do affect the position of the graph
on the coordinate plane though).
Below are two examples of equations that are circles. Give two more examples. Remember that A and C (the
coefficients of x2 and y2) must be the same.
1. 5x2 + 5y2 = 15
2. 4x2 +3x + 4y2 -15 = 32
3. ___________________________
4. ____________________________________
Consider the next set of equations:
Equation
3x2 + 5y2 = 4
2x2 + 7y2 = 8
- 3x2 - 8y2 = 4
4x2 = 18 - 9y2
Graph
ellipse
ellipse
ellipse
ellipse
What do all of these equations that produce an ellipse as a graph have in common (look the coefficient of the
x2 and y2 terms when the equation is in general form)?
______________________________________________________________________________________
You are correct if you noticed that the coefficients of x2 and y2 in each equation have the same sign but are
not equal. So any equation of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 in which A and C have the same
sign but are not equal will produce an ellipse when graphed. The values B, D, E, and F will not affect the
general shape (they do affect the position of the graph on the coordinate plane though).
Below are two examples of equations that are ellipses. Give two more examples. Remember that A and C
(the coefficients of x2 and y2) must be the same sign but different in value.
1. 5x2 + 9y2 = 10
2. x2 - 3x + 4y2 + 30 = 40
3. ___________________________
4. ____________________________________
2
Now consider:
Equation
3x2 - 5y2 = 4
2x2 - 4y2 = 8
- 3x2 + 8y2 = 4
6x2 = 18 + 9y2
Graph
hyperbola
hyperbola
hyperbola
hyperbola
What do all of these equations that produce a hyperbola as a graph have in common (look the coefficient of
the x2 and y2 terms when the equation is in general form)?
______________________________________________________________________________________
You are correct - if you noticed that the coefficients of x2 and y2 have different signs. So any equation of the
form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 in which A and C have different signs will produce a hyperbola
when graphed. The values B, D, E, and F will not affect the general shape (they do affect the position of the
graph on the coordinate plane though).
Below are two examples of equations that are hyperbolas. Give two more examples. Remember that A and C
(the coefficients of x2 and y2) must be different in sign.
1. 5x2 - 9y2 = 10
2. x2 - 3x - 4y2 + 30 = 40
3. ___________________________
4. ____________________________________
SUMMARY OF THE GENERAL FORM AND GRAPHS:
For the general conic section: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
circle
ellipse
hyperbola
A=C
A ≠ C but both positive or both negative
A and C are different in sign
The values B, D, E and F further define the characteristics of the graphs of the conic section (such as the
diameter of the circle, length of the ellipse etc) as well as the position of the graph on the coordinate plane.
PROBLEM SET #1 (answers at the end of this worksheet)
Identify the general shape of the graph produced by each of the following equations. Remember that the
equation must be put into general form (zero is on one side of the equation) before you can determine what
graph will be produced.
1. x2 - 3xy + 5y2 -2 =30
2. 4x2 - 5x = 6y2 - 6y + 10
3. 5x - 3y = 15
4. 6x2 + 6y2 = 29
5. 3x2 - 4y + 5y2 - 34 = 0
6. 8x2 - 4y + x = 18
7. 8x2 - 3y - 18 = 3y2
8. x2 + 3xy = 4y - y2 + 36
9. y = 3x2 - 5x + 1
10. 8x2 + 5x + y2 -2x = 23
3
WHERE DO CONIC SECTIONS COME FROM???
In this section, we give geometric definitions of circles, parabolas, ellipse and hyperbolas and will state their
standard equations. These figures are called conic sections, or conics, because they result from intersecting a
cone(s) with a plane, as shown below. For an animated look at how these figures are formed, visit the web
site: http://chuwm2.tripod.com/conics/flash.htm (you may need to temporarily allow pop-ups from this site).
CIRCLES
A circle is the collection of points (x, y) in the plane with the property that the distance from any of these
points to a fixed point (h, k) is a constant length, r.
The point (h, k) is the center of the circle and the constant, r, is the radius.
(h, k)
The equation of the circle to the right in standard form is:
r
(x - h)2 + (y - k)2 = r2
(x, y)
Finding the center and radius.
If the equation of a circle is in standard form, it’s easy to find the center and radius
by picking the values of h, k and r from the equation.
4
Here’s an example.
(2, 3)
2
2
2
Equation: (x - 2) + (y - 3) = 4
Center: (2, 3)
Radius: 4
The graph of this circle is on the right. Notice the same information
pulled from the equation in standard form can also be obtained from the graph itself by determining the
coordinates of the center and then counting the number of units to the edge of the graph (radius).
4
Writing the equation and sketching the graph.
If we know the center and the radius of a circle, we can find it’s equation or sketch its graph.
In the circle on the right, we know the center is at (0, -1) and the radius is 3
(count from the center vertically or horizontally to the edge of the circle).
Here’s how we can find the equation.
Start with the equation of a circle with
center (h, k) and radius r.
Substitute 0 for h and -1 for k
and 3 for r:
3
(0, -1)
(x - h)2 + (y - k)2 = r2
(x - 0)2 + (y - (-1))2 = 32
Simplify, being careful to make the sign change:
x2 + (y + 1)2 = 9
PROBLEM SET #2: (answers at the end of this worksheet)
Use the standard form of the equation of a circle (x - h)2 + (y - k)2 = r2 to answer each of the following:
1. What is the center and radius of the circle whose equation is:
a. (x - 3)2 + (y - 4)2 = 16
b. (x + 2)2 + (y - 5)2 = 20
c. (x - 1)2 + (y + 6)2 = 36
d. x2 + y2 = 18
e. (x - 3)2 + y2 = 25
2. Write the equation of the circle:
a. with center (-1, 2) and radius 5
b. with center (0, 0) and radius 3.2
c. with center (3, -5) and radius 2
d. whose graph A. is shown at the right
e. whose graph B. is shown at the right
3. Sketch the graph of the circles in question #1 in this problem set
5
PARABOLAS
Parabolas may open up, down, to the right, or to the left. We will concern ourselves only with the parabolas
that open up or down.
In general, a parabola is the collection of points (x, y) in the plane that are the same distance from a fixed
line (called the directrix) and a fixed point (called the focus).
The point (h, k) on the parabola is the vertex and
it lies halfway between the focus and the directrix.
A parabola also has an axis of symmetry that
passes though the focus and vertex.
(h, k)
Here’s the standard form of the equation
of a parabola that opens up (or down):
(x - h)2 = 4a (y - k)
where (h, k) is the vertex and a is the value which determines the width of the opening of the parabola as
well as whether it opens up (a is positive) or open down (a is negative).
Writing the equation.
If we know the vertex (h, k) and a point on the graph (x, y), we can write the equation of the parabola.
For example, write the equation of the parabola whose vertex is (5, -1) and passes through the point (7, 3).
Begin with the standard form of the parabola:
(x - h)2 = 4a (y - k)
Substitute 5 for h, -1 for k, 7 for x and 3 for y:
(7 - 5)2 = 4a (3 - ( -1))
Simplify, being careful with the sign change
(2)2 = 4a(3 + 1)
4 = 4a ● 4
4 = 16a
4
= a or a = 0.25
16
The equation would then be: (x - 5)2 = 4(0.25) (y - -1) or (x - 5)2 = 1 (y + 1)
PROBLEM SET #3: (answers at the end of this worksheet)
Write the equation of each of the following parabolas:
1. The vertex is located at (2, 4) and the point (1, 5) is on the parabola (you will need to use the standard
form of a parabola opening up or down (x - h)2 = 4a (y - k))
2. The vertex is located at (0, 0) and the point (2, - 6) is on the parabola (standard form of a parabola opening
up or down: (x - h)2 = 4a (y - k))
6
ELLIPSES
An ellipse is the collection of all points (x, y) in the plane with the property that the sum of the distances (d1
+ d2) from (x, y) to two fixed points (called foci) is constant.
b
a
d1
d2
The standard form for the equation of this ellipse is:
x2 y2
+
=1
a 2 b2
where: the center of the ellipse is at the origin (0, 0)
the major axis is along the x-axis and has length 2a
the minor axis is along the y-axis and has length 2b
x2 y2
+
=1
32 5 2
the center of the ellipse is at the origin (0, 0)
the major axis has length 6 (two times 3)
the minor axis has length 10 (two times 5)
Ex: if the equation of an ellipse is:
then:
Writing the equation and sketching the graph.
Writing the equation and sketching the graph are easy when the equation is in standard form.
To sketch a graph of the ellipse:
1. locate the center of the ellipse (it will always be at the origin at this point in your studies)
2. locate a point a units left and a point a units right to locate the horizontal width of the ellipse
3. locate a point b units up and a point b units down from the center to locate the vertical height of
the ellipse
4. Use a smooth curve to connect the four points you have plotted to sketch the ellipse.
For example, sketch the graph of
x2 y2
+
=1
22 42
1. locate the center of the ellipse at the origin
2. locate a point 2 units left and a point 2 units right of the origin
3. locate a point 4 units up and a point 4 units down from the origin
4. Connect the points to finish the ellipse
4
2
2
4
x2 y2
+
= 1 , you would have to
4 16
determine that a is 2 and b is 4 by taking the square roots of 4 and 16.
Note: if the equation was simplified as
7
Similarly, it is also easy to write the equation if the graph is give. The value for a can be found by counting.
Start at the origin and count the number of units to the right OR left to the graph. This is the value of a. The
value of b can be found by counting the number of units from the center up OR down to the graph.
Once you know these values, substitute them into the standard form of the ellipse.
For example, write the equation of the ellipse show at the right.
Count the units from the center to the right or left side: 3 units
Count the units from the center to the top or bottom: 2 units
Now substitute 3 for a and 2 for b into the
standard form of the equation of an ellipse:
and simplify:
x2 y2
+
=1
32 2 2
x2 y2
+
=1
9
4
PROBLEM SET #4: (answers at the end of this worksheet)
Sketch the graph of each of the following:
x2 y 2
x2 y 2
+
=1
+
=1
2.
1.
2 2 32
16 25
x2 y 2
3.
+
=1
5 10
Write the equation of each of the following:
4.
5.
6.
8
HYPERBOLA
Hyperbolas may open left and right or may open up and down. We will concern ourselves only with
hyperbolas that open right and left.
A hyperbola is the collection of all points (x, y) in the plane with the property that the difference of the
distances (the absolute value of the difference d1 - d2) from (x, y) to two fixed points (foci) is a positive
constant.
focus
focus
The standard form of the equation of this hyperbola is:
x2 y 2
−
= 1 where
a 2 b2
the center is at the origin (0, 0)
the vertices are at the points (-a, 0) and (a, 0)
the transverse axis is along the x-axis and has length 2a
Example: if the equation of the hyperbola is:
x2 y2
−
=1
2 2 52
then the center is at the origin (0, 0)
the vertices are at the points (-2, 0) and (2, 0)
the transverse axis is along the x-axis and has length 5
Writing the equation and sketching the graph.
To graph a hyperbola, it is easiest to first draw the fundamental rectangle which helps locate the arms of the
hyperbola.
To do this,
1. locate the center (0, 0) in this case (the center is not always at the origin but for all of our cases it will be).
2. from the center move a units right and then a units left. The points you have located determine the left and
right sides of the fundamental rectangle. Draw vertical lines through each of these points.
3. from the center move b units up and b units down. The points you have located determine the top and
bottom sides of the fundamental rectangle. Draw horizontal lines through each of these points.
4. you now have the fundamental rectangle. Draw the diagonals of the rectangle and extend them. These
extended diagonals are the asymptotes of the hyperbola (the hyperbola approaches but does not touch the
asymptotes). Using them as guides, sketch the hyperbola.
9
For example, sketch the graph of the hyperbola:
x2 y2
−
=1
22 42
1
1. the center is located at (0, 0) so from there locate a point 2 units left and a point 2 units right of the center;
locate a point 4 units up and a point 4 units down from the center.
2. draw the sides of the fundamental rectangle using your plotted points as the guide.
3. draw the diagonal of the fundamental rectangle, extending them beyond the rectangle itself.
step 2
step 1
2
step 3
4
4
4. Using the asymptotes as a guide, sketch the hyperbola. It will pass though the points on the x-axis and the
arms will approach but not touch the diagonals of the fundamental rectangle.
Writing the equation a reverse process. The values for a and b must be determined by counting from the
center of the hyperbola to the top and bottom and then left and right of the fundamental rectangle. These are
the values that will replace a and b in the equation.
For example, in the graph to the right, the value of a is 4 and
the value for b is 1. The equation then would be
x2 y2
x2 y2
−
= 1 or simplified:
−
=1
16 1
4 2 12
10
PROBLEM SET #5: (answers at the end of this worksheet)
Sketch the graph of each of the following hyperbolas using the fundamental rectangle and it diagonals.
x2 y 2
x2 y2
x2 y2
2.
1. 2 − 2 = 1
−
=1
3.
−
=1
16 25
8 20
3
2
Write the equation of each of the hyperbolas shown below.
4.
5.
6.
Summary of what you need to know about conic sections.
1. Identify the shape of the graph given its equation in GENERAL form
2. Identify the shape of the graph given its equation in STANDARD form
3. Sketch an ellipse, hyperbola or circle given its equation in standard form
4. Write the equation of an ellipse, hyperbola or circle given its graph
5. Write the equation of a parabola given the vertex and one point on the graph.
You have now completed Topic 15.1. Homework problems are on page 14.
11
ANSWERS
Problem Set #1
1. ellipse
2. hyperbola
6. parabola
7. hyperbola
3. line
8. circle
Problem Set #2
1. a. (3, 4) , r = 4
d. (0, 0). r = 18
b. (- 2, 5), r = 20
e. (3, 0), r = 5
c. (1, - 6), r = 6
2. a. (x + 1)2 + (y - 2)2 = 25
d. (x + 1)2 + y2 = 9
b. x2 + y2 = 10.24
e. (x - 1)2 + (y - 1)2 = 16
c. (x - 3)2 + (y + 5)2 = 4
3. a.
b.
d.
e.
Problem Set #3
1. (x - 2)2 = 4•1/4 (y - 4)
or (x - 2)2 = (y - 4)
4. circle
9. parabola
5. ellipse
10. ellipse
c.
2. (x - 0)2 = 4•(-1/6) (y - 0)
or y = -3/2 x2
12
Problem Set #4
1.
4.
x2 y2
+
=1
16
4
2.
5.
3.
x2 y 2
+
=1
5
1
6.
x2 y 2
+
=1
16 25
Problem Set #5
1.
2.
3.
units are 2
units are 2
4.
x2 y2
−
=1
4
9
5.
x2 y 2
−
=1
1 16
6.
x2 y2
−
=1
4
4
13
HOMEWORK QUESTIONS FOR CONIC SECTIONS
1. Write the equation in standard form of the circle whose center is at (- 4, 1) that has radius 10 units.
2. Graph the circle whose equation is
(x + 5)2 + (y - 1)2 = 16
3. Write the equation of in standard form of the circle whose center is at (1, 1) that passes through the point
(4, 5).
5. Graph the circle whose equation is
(x - 2)2 + (y + 1)2 = 12
6. Graph the parabola whose equation is
(x - 1)2 = 4(y + 1)
7. The cross section of a tunnel is a semicircle of radius 35 feet. Will a truck that is 15 feet tall be able to
make it thru the tunnel if it drives 30 feet from the center of the tunnel? (Hint: use the figure below and the
equation x2 + y2 = (35)2 to find the height of the tunnel, y, at 30 feet from the center.)
(30, Y)
(0, 0)
(30, 0)
(35, 0)
8. Graph the ellipse whose equation is
x2 y2
x2 y2
+
=1
b.
+
=1
a.
16 9
49 16
9. Graph the hyperbola whose equation is
y2 x2
x2 y2
a.
−
=1
b.
−
=1
36 4
4
1
c.
x2 y2
−
=1
16 4
10. Given two equation of two hyperbolas:
x2 y2
−
=1
9
4
and
x2 y 2
−
=1
25 16
Determine if the graphs of the two equations intersect. (Hint: sketch the two graphs on the same coordinate
system!)
14
Homework Answers
1. (x + 4)2 + (y - 1)2 = 102
2.
3. (x - 1)2 + (y - 1)2 = 25
5.
6.
7. Yes, the truck will make is since the truck is 15 feet tall but the bridge is about 18 feet tall at 30 feet.
8. a.
15
8 b.
Units on graph is 2
9 a.
x unit is 2, y unit is 1
9.b.
9. c.
units on graph are 1
10. Yes (sketch the graphs on the same coordinate plane)
16