Lesson Identify elements of a hyperbola given by its general equation

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Identify elements of a hyperbola given by its general equation

This lesson teaches you by examples on how to identify the center, vertices and foci of a hyperbola given by its general equation.
Prerequisites to this lesson are the lessons
    - Hyperbola definition, canonical equation, characteristic points and elements
    - Standard equation of a hyperbola
    - General equation of a hyperbola
    - Transform general equation of a hyperbola to the standard form by completing the square
in this site.

Problem 1

Find the standard equation, verices and foci of the hyperbola 25x%5E2-39y%5E2%2B150x%2B390y

.

Solution

 =   --->    (we need to complete squares for x and y separately)  ---> 

 =        (so we group the x-terms and y-terms separately)

 =          (preparing to complete squares)

 =     (completing the squares)

 =             (completing the squares is just done)

 =              (finalizing completing the squares)
 
  
   
 = 



The standard equation above tells us that the center is at (-3,5) ;

the real (or transverse) axis is x= -3 (vertical line parallel to y-axis);           

the linear eccentricity is  =  =  = 8; 

Since the foci are on the real axis at a distance c = 8 from the center,

their coordinates are  (-3,5+8) = (-3,1)  and  (-3,5-8) = (-3,-3).

As for the vertices, they also on the real axis,
and their coordinates are  (-3,5+5) = (-3,10)  and  (-3,5-5) = (-3,0).

  
   



  Hyperbola  -  = 1