Translation of Axes

Previously, we derived equations for the conic sections
     parabola
     ellipse
     hyperbola
when their axes coincided with the x and y axes of the Cartesian coordinate system
and centered relative to the origin of the Cartesian coordinate axes
Now we will look at the equations of the conic sections when the conic sections
are moved away from the origin of the Cartesian coordinate axes while the axes
of the conic sections remain parallel
to the x and y axes of the Cartesian coordinate system

In other words, we move the conic sections using only parallel translation motions
We will not in any way rotate the conic sections

The equations we will get for these parallel translated conic sections all will look like
          
where A and C are not both zero


               Translation of Axes

A translation of coordinate axes occurs when the new coordinate axes
have the same direction as and are parallel to the original coordinate axes

               

A point P on the plane has two sets of coordinates
                                              (x,y)
in the original coordinate system
                                             (x’,y’)
in the translated coordinate system



                  Translation Formulas

                                      
                                               or
                                    


             See Example 1, page 884, of the textbook




               Standard Equations of Translated Conic Sections

                                                 Parabolas

               




               



                                                         Circle

               



                                             Ellipses


               



               




                                               Hyperbolas


               




               



          Graphing Translated Conic Sections

The graph of an equations of the form
          
where A and C are not both zero is
     a conic section
or
     a degenerate conic section
or
     there is no graph

If we can transform the equation into one of the standard forms,
then we can quickly identify its graph and sketch it

The method of transforming the equation is the process of completing the squares

     See Examples 2 – 3, pages 887 – 889, of the textbook


          Finding Equations of Conic Sections
Given certain information about a conic section in rectangular coordinates,
we can find its equation

     See Example 4, pages 890 – 891, of the textbook


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