Derive the standard equation of the ellipse: x^2/b^2+ y^2/a^2=1
An ellipse is the locus of points in a plane whose sum of the distances from two fixed points is a constant.
The two fixed points are the foci, `F_1,F_2` . The constant distance is represented by `2a` .
(1) Without loss of generality, let `F_1` be located in the Cartesian plane at (-c,0), and `F_2` at (c,0).
(2) Consider a point (x,y) on the ellipse. The distance from (x,y) to `F_1` is `r_1` , the distance from (x,y) to `F_2` is `r_2` , and we have :
`r_1+r_2=2a` for any point (x,y) on the ellipse.
(3) `r_1=sqrt((x+c)^2+y^2)` and `r_2=sqrt((x-c)^2+y^2)` . Then
`x^2/a^2+y^2/(a^2-c^2)=1` Define `b^2=a^2-c^2`
Where 2a is the length of the major axis, 2b the length of the minor axis, and c the distance between the foci. (Notice b is defined in terms of the semi-major axis and the focal length)
This is the equation of an ellipse in standard form, centered at the origin.
The given equation: x^2/b^2+ y^2/a^2=1 is in the standard form already. It is not possible to plot the graph of this ellipse until the value of a and b is known.
If a>b, the ellipse has a horizontal major axis and if a<b the ellipse has a vertical major axis
Taking random values of a = 2 and b = 1, the graph of the resulting ellipse is