Standard Form of Ellipse

Standard form of Ellipse:
             Ellipse is the two dimensional closed geometric figure formed by the intersection of the circular cone and the plane cutting through the circular cone completely. The distance between any point on the ellipse and the foci is always constant one. The addition of distance between any two fixed points is constant

standard formula

The standard form of the ellipse is
                       ` (x^2/m^2)` +`(y^2/n^2)` = 1 where m and n are greater than 0
                       Here
                                  Origin is (0, 0)
                                  The length of the major axis is 2m
                                  The length of the minor axis is 2n

Model problems for standard form of ellipse

1.The equation of the ellipse is
                         4x²+9y²=36
a.Find the x intercept and y intercept.
b.Find the points of the foci
c.Find the length of the major and minor axis
d.Draw the graph
Solution:
          Here the standard equation is
                   The standard form of the ellipse is
                       ` (x^2/m^2)` +`(y^2/n^2)` = 1 where m and n are greater than 0
                       Here
                                  Origin is (0, 0)
                                  The length of the major axis is 2m
                                  The length of the minor axis is 2n
Rewrite the equation in the standard form so, divide the entire equation by 36
              ` ((4x^2)/36)` + `((9y^2)/36)` = `(36/36)`
                        `(x^2/9)` +`(y^2/4)` = 1
                    Here m²= 9, m=3
                            n² =4,n=2             m>n>0
^{a.Find the x intercept and y intercept:}
   To find the x intercept put y=0 in the given  equation
                                 ` (x^2/9)` +0=1
                                         x²=1*9
                                         x²= 9
                                            x= + or – 3
   To find the y intercept put x=0 in the given equation
                                0 +`(y^2/4)` =1
                                            y²=1*4
                                             y²=4
                                               y= + or – 2


b.Find the point of foci:
                     We need to find the p
                                           p²= 9-4
                                           p²=5
                                              p = + or – 2.23
       The foci points are (2.23, 0) and (-2.23, 0)
c.The length of the major axis and minor axis
           The length of the major axis is 2m = 2*3= 6
           The length of the minor axis is 2n = 2*2=4
d.We need to draw the graph