Name
June 2, 2014
Honors Advanced Mathematics notes and problems
Rotating ellipses and hyperbolas page 1
Rotating ellipses and hyperbolas
Below are general procedures for rotating ellipses and hyperbolas counterclockwise by angle . In class we
will do examples using specific  values, which may make the process easier to understand. Also note that
while our current interest is rotating ellipses and hyperbolas, the same techniques would work for rotating a
graph of any kind of function or relation. To illustrate, the last problem in the homework asks you to rotate
a parabola, and these same methods will apply.
Using rectangular equations
Step 1: Write a transformation equation description of the rotation.
cos( )  sin( )  x   x 
 sin( ) cos( )   y    y 

   
Step 2: Solve for the input variables x and y (which involves finding an inverse matrix).
1
 x  cos( )  sin( )  x  cos( ) sin( )   x 
 y    sin( ) cos( )   y    sin( ) cos( )  y 
  
   
 
Step 3: Rewrite the matrix equation as separate statements for x and y.
x = cos( ) x  sin( ) y 
y =  sin( ) x  cos( ) y 
Step 4: Make substitutions into the equation for the ellipse or hyperbola, replacing x and y with the
expressions from the previous step.
Step 5: Having finished applying the transformation, change x  and y  to plain x and y, and simplify the
answer as much as possible.
Using parametric equations
Finding parametric equations for rotated curves is considerably easier, because the transformation matrix
can be applied directly to the expressions from the parametric equations, as shown below. There’s no need
to solve the input variables so an inverse matrix isn’t needed. Here’s a description of how to do it for an
ellipse, but the parametric equations for the ellipse in Step 2 could be replaced by parametric equations for
any other type of curve.
Step 1: Write a transformation equation description of the transformation.
cos( )  sin( )  x   x 
 sin( ) cos( )   y    y 

   
Step 2: Replace x and y with parametric formulas for the ellipse.
cos( )  sin( ) (rx cos(t )  h)  x
 sin( ) cos( )   (r sin(t )  k )    y 

 y
  
Step 3: Do the matrix multiplication, and write the result in a non-matrix form.
x  cos( )(rx cos(t )  h)  sin( )(ry sin(t )  k )
y   sin( )(rx cos(t )  h)  cos( )(ry sin(t )  k )
Step 4: Having finished applying the transformation, change x  and y  to plain x and y, and simplify the
answer as much as possible.
Name
June 2, 2014
Honors Advanced Mathematics notes and problems
Rotating ellipses and hyperbolas page 2
Problems
1. Start with the ellipse centered at the origin, having (±3, 0) and (0, ±8) as endpoints of its axes. Now
consider rotating by 4 clockwise (use  = – 4 ) around the origin to get a new ellipse.
a. Write a rectangular equation for the original ellipse. Following the process outlined on the top of
page 1, find a rectangular equation for the rotated ellipse. Simplify your answer.
b. Graph both the original ellipse and the rotated ellipse on your calculator in ordinary function
mode. (Before doing so, you will need to solve both equations for y. For the rotated ellipse, the
quadratic formula helps you solve for y.) Make sure the graphs look as expected. Sketch them on
paper.
c. Write parametric equations for the original ellipse. Following the process outlined on the bottom of
page 1, find parametric equations for the rotated ellipse. Simplify your answer.
d. Graph both the original ellipse and the rotated ellipse on your calculator in parametric mode. Make
sure the graphs look as expected, and agree with the graphs you saw in part b.
2. a–d. Repeat problem 1 with the following specifications: original ellipse centered at (2, –3) having a
horizontal major axis of length 12 and a vertical minor axis of length 4, and a rotation angle of
120 counterclockwise.
e. What was the center of rotation in this problem: the origin (0, 0) or the ellipse’s center (2, –3)?
3. Start with the hyperbola  4x  
origin to get a hyperbola.
2

y 2
6
 1 , and consider rotating by

3
counterclockwise around the
a. Find a rectangular equation for the rotated hyperbola. Simplify your answer.
b. Graph both the original hyperbola and the rotated hyperbola on your calculator in ordinary
function mode. (Before doing so, you will need to solve both equations for y. For the rotated
hyperbola, remember that the quadratic formula helps.) Make sure the graphs look as expected.
Sketch them on paper.
c. Remember from last week, that he parametric equations for this hyperbola are x = 4 sec t, y = 6 tan.
Following the process outlined on the bottom of page 1, find parametric equations for the rotated
hyperbola. Simplify your answer.
d. Graph both the original hyperbola and the rotated hyperbola on your calculator in parametric mode.
Make sure the graphs look as expected, and agree with the graphs you saw in part b.
4. Repeat all parts of problem 3 but using 2 clockwise ( = – 2 ) as the rotation angle. This time the
resulting hyperbola will have its axes running vertically and horizontally, so when you’re simplifying
in part a, make the equation fits the standard form you’ve learned for such hyperbolas.

5. Start with the hyperbola  cx   cy  1 , and rotate it by 4 counterclockwise. Simplify your answer
into the form xy = k where k will be some constant expression involving c.
(Notice that this proves that equations of the form xy = k have hyperbola graphs.)
2
2
6. The method you used for rotating ellipses and hyperbolas works also for other types of curves, such as
parabolas. Using the method from the top half of page 1, rotate the parabola y = x2 by

3 counterclockwise. Simplify the resulting equation as much as you can.