.well-known

This is a Brave Rewards publisher verification file. Domain: mathematicsway.blogspot.com Token: 61623a436e7c44ba1310544150d3db370336fd3a7242d9db2869ad9e33470742

Followers

Friday, February 27, 2015

ELLIPSE and HYPERBOLA

ELLIPSE

        An ellipse is a locus of a point which moves so that the sum of its distances from two fixed
     points is constant. The fixed points are called the foci of the ellipse and the line joining them
     is the principal axis.  
   An ellipse is symmetric with respect to its principal axis ( F’F ) and also with respect to the line B’B which is the perpendicular bisector of  F’F. The point of intersection O of B’B and F’F  is thus a point of symmetry and is called the center of the ellipse.


                      
                       This is the standard equation of an ellipse with center at the origin and

                             ( x – h )2   +   ( y – k )2     =  1         for ellipse with center at the point  (h, k). 
                                 a2                    b2

           where,  a  is the length of the semimajor axis and  b  is the length of the semiminor axis.

              If  a  >  b , the axis of the ellipse is horizontal and  if  a < b ,  the axis is vertical.
 






                c  =  √ a2 – b2   =  the distance from the center of the ellipse to the foci = CF = CF’

                b  = the distance from the center of the ellipse to the minor vertices  = CB = CB’

                a  =  the distance from the center of the ellipse to the major vertices  = CV = CV’


          The equation  Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 is a general equation of an ellipse if 
           B = 0  and A  ¹  C and  both A and C have the same sign.


                Example
                     Find the center, vertices and foci of the given ellipse.

                       1.   x2  +  4y2  =  16                                 a > b , axis is horizontal

                             x2  +  4y2  = 16                                  a =  4 ,   b = 2  and      c = √ 12    
         
                             x2   +    y2   =  1   .                             center, C( 0 , 0 )      
                            16          4                                           V( 4, 0 )  and  V’ (– 4 , 0 )

                       ( x – 0 ) 2   +   ( y – 0 )2   =  1   .              B ( 0 , 2 )  and  B’( 0 , – 2)
                   42                    22                              F (√ 12 ,  0 )   and  F’ (– √ 12 ,   0 )

                                                      Graph of the ellipse   x2  +  4y2  =  16  

                                               

                   Do these exercise.  Find the center, foci, major and minor vertices of the following ellipse
                    and draw the graph.

                                   1.    x2  +  4y2 – 8y  =  12                              

                                   2.    x2  +  4y2 – 6x  =  7                                

                                   3.    x2  +  4y2 – 4x – 8y  =  0           
             
                                   4.    9x2  +  4y2  =  36

                                   5.   16x2  + 25y2 – 128x – 150y + 381 = 0

                                   6.   25x2  + 16y2 – 100x – 32y  =  284



                     HYPERBOLA
           A hyperbola is  locus of point which moves so that the difference of its distances from two fixed
           points is constant. The fixed points are called the foci of the hyperbola and the line joining them is
          the principal axis. The segment V’V is the transverse axis having length of 2a and the segment B’B
         is the conjugate axis whose length is 2b. The segment F’F is the principal axis and is equal to 2c.
           
                                                                   c2  =  a2  +  b2 

                      Equation of Hyperbola      

                            The general equation of  hyperbola is defined by

                                                    Ax2 + Cy2 + Dx + Ey + F = 0

                                                    where  A and C have unlike signs.  


                            The standard equation of a hyperbola is given by

                                      a) with C ( 0 , 0 )

                         x 2    –     y 2    =  1           For hyperbola with horizontal transverse axis.
                        a2            b2

                          y 2    –     x 2    =  1           For hyperbola with vertical transverse axis.
                     a2            b2

                        b) with C ( h , k )

                          ( x – h ) 2   –   ( y – k ) 2   =  1     For hyperbola with horizontal transverse axis.
                         a2                    b2

                      ( y – k ) 2   –   ( x – h ) 2   =  1      For hyperbola with vertical transverse axis.
                               a2                   b2


                          Example 1.    x2  –  4y2  =  16                                           
              

                              Solution :
                          
                                          x2  –  4y2  = 16                                               
         
                                            x2    –    4y2     =    16                                      
                                           16           16           16

                                             x2    –     y2    =  1                                                    
                                            16           4          

                                     ( x – 0 )2    –   ( y – 0 )2     =  1                           
                                           42                    22                              
                                     a =  4 ,   b = 2  and    c = √ 16  +  4     =  √20 


                                     C ( 0 , 0 )  ,   axis is horizontal


                                    V ( 4 , 0 ) ,  V’( – 4 , 0 ) ,    
                      
                                     F (√20 , 0 ) ,  F’(–√20 , 0 ) , 

                                    B( 0 , 2 ) ,  B’( 0 , – 2)     


              Do these exercises. 
                Find the center, conjugate and transverse vertices,  foci and asymptote of the given hyperbola
                and draw the graph.

                               1.    x2  –  4y2 – 8y  =  12                             

                               2.    y2  –  4x2  +  6x  =  7                             

                               3.  x2  –  4y2 – 4x – 8y  =  0                         

                              4.   4x2  –  9y2  =  36

                              5.  16y2  – 25y2 – 128x – 150y + 381 = 0

                              6.   16y2  –   25x2  – 100x – 32y  =  284

                              7.   9x2  –  4y2   =  36

                              8.   9x2  –  18x  –  4y2  +  8y  =  31
 
                              9.   4x2  –  9y2  + 18y   =  45

                            10.   9x2  +  18x  –  4y2   =  27 



No comments:

Post a Comment