Derive the standard equation of the ellipse:  x^2/b^2+ y^2/a^2=1

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

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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

`sqrt((x+c)^2+y^2)+sqrt((x-c)^2+y^2)=2a`

`sqrt((x+c)^2+y^2)=2a-sqrt((x-c)^2+y^2)`

`(x+c)^2+y^2=4a^2-4asqrt((x-c)^2+y^2)+(x-c)^2+y^2`

`4cx-4a^2=-4a sqrt((x-c)^2+y^2)`

`a-c/ax=sqrt((x-c)^2+y^2)`

`a^2-2a c/ax+(c^2)/(a^2)x^2=x^2-2cx+c^2+y^2`

`x^2((a^2-c^2)/a^2)+y^2=a^2-c^2`

`x^2/a^2+y^2/(a^2-c^2)=1` Define `b^2=a^2-c^2`

`x^2/a^2+y^2/b^2=1`

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.

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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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

:

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