Name Worksheet 13.4 Parabolas, Directrix, Eccentricity 1. Focus

Name _____________________
Worksheet 13.4
Parabolas, Directrix, Eccentricity
1. Focus/Directrix Definition of a _______________________
Every Conic Section can be defined in terms of a single focus and a directrix. A focus is a fixed
point, while a directrix is any fixed line (that usually does not pass through the focus).
(a) Use the “single bullet” paper to draw the locus of points that are equidistant from a point (the
center of the concentric circles) and one of the many straight lines. Choose a different line
from the other people at your table.
(b) Which conic section have you created?
(c) Use the distance formula to derive the equation for this conic section.
2. Use the distance formula to find a Cartesian equation for the following parabolas. Use
algebra to rearrange your equation into the standard form for parabolas:


( x  h) 2  a( y  k ) , if the parabola opens up or down, where a is some constant.
( y  k )2  a( x  h) , if the parabola opens right or left, where a is some constant.
Your class should distribute the work of finding the Cartesian equation for each of the
following 10 parabolas. Summarize the class results in the table on the following page.
These parabolas open up/down
a. Focus (0, 1) and Directrix y = −1.
b. Focus (0, 2) and Directrix y = −2.
c. Focus (0, −1) and Directrix y = 1.
P (x,y)
Focus
d. Focus (0, 1) and Directrix y = 0.
e. Focus (0, −2) and Directrix y = 1.
Directrix
These parabolas open left/right
f. Focus (1, 0) and Directrix x = −1.
g. Focus (−2, 0) and Directrix x = 0.
P (x,y)
h. Focus (−2, 0) and Directrix x = 1.
i. Focus (0, 0) and Directrix x = −3.
Focus
j. Focus (−2, 0) and Directrix x = 2.
Directrix
Parabola
Coordinates
(h, k) of
Vertex
Distance from
Focus to
Vertex
Coordinates
(h, k) of
Vertex
Distance from
Focus to
Vertex
Cartesian Equation
Value of a
( x  h) 2  a( y  k )
a. Focus (0, 1) and Directrix y = −1.
b. Focus (0, 2) and Directrix y = −2.
c. Focus (0, −1) and Directrix y = 1.
d. Focus (0, 1) and Directrix y = 0.
e. Focus (0, −2) and Directrix y = 1.
Parabola
Cartesian Equation
Value of a
( y  k )  a( x  h)
2
f. Focus (1, 0) and Directrix x = −1.
g. Focus (−2, 0) and Drectrix x = 0.
h. Focus (−2, 0) and Directrix x = 1.
i. Focus (0, 0) and Directrix x = −3.
j. Focus (−2, 0) and Directrix x = 2.
3. Using your chart above, how is a related to the distance from the vertex to the focus. Draw a
picture to help you remember. What does it mean if the value of a is negative?
4. Given the equations of the parabolas below, determine the location of the vertex, focus, and
directrix. Make a quick sketch of each parabola.
a. x 2  2( y  1)
c.
1 2
x  ( y  1)
4
b. y  6( x  1) 2
d. x  1  8( y  3)2
Hint on the following exercises: Complete the square and rearrange into standard form.
d. 0  2 y 2  4 x  4 y  12
e. 0  x2  2 y  4 x  8
5. Focus/Directrix Definition of a ________________________
(a) Using the “single bullet” paper, take the focus F  (0, 3) as the center of the circles and the
d ( F , P) 1
line y  12 as the directrix. Draw various points P such that
 .
d ( D, P ) 2
(b) Describe the curve you have found. Is it a closed curve? Where is the center of the figure?
(c) Give the coordinates of any points you know that are certainly on the curve as well as the
coordinates of the foci (foci corresponding to the focus/focus definition of the same conic
section).
(d) Find the equation that describes this conic section.
6. Consider the ellipse given by
x2 y 2

1
16 25
(a) Sketch.
(b) Find the coordinates of both foci.
(c) Find the eccentricity of the ellipse.
(d) This same ellipse could be defined by a directrix and the focus (0, c) -- one of your two
answers to part b. Find the equation for this directrix.
7. Focus/Directrix Definition of a ____________________
Consider F1 as the single focus and D as the point on the directrix closest to a point P. The locus
d ( F , P)
of all points such that
 e creates a conic section. This constant e is a constant for each
d ( D, P )
conic called the eccentricity of the conic section.
(a) Using the “single bullet” paper, take the focus F  (0, 0) as the center of the circles and the
d ( F , P) 3
line y  5 as the directrix. Draw various points P such that
 (D is the point on
d ( D, P ) 2
the directrix closest to P).
(b) Describe the curve you have drawn. Is it a closed curve?
(c) Use the distance formula to find an equation that describes this locus of points.
2
2
8. Consider the hyperbola x  y  1
16 9
(a) Rearrange the equation to read y  f ( x) . Draw the curve on your calculator in function
(normal) mode.
(b) Use this rearrangement (what happens as x gets really big?) to help you find the equations of
the two asymptotes. What are the two asymptotes? Make a sketch!
(c) Find the two foci of this hyperbola.
(d) Find the eccentricity.
(e) Find the equation for a directrix that could also define this same hyperbola (along with one of
the two foci).