equation of ellipse

Let us learn about equation of ellipse

Equation of ellipse in standard form

[ x²/a² ] + [ y²/b² ] = 1

Let x'ox & yoy' be the co-ordinate axes.

Let F(c, o) and f'(- c, o) be 2 given fixed points .

Let us consider the locus of a point which moves in such a way that the sum of its distances from F & F' remains constant say equal to 2a where a > c.

Let P(x, y) be any point on the locus.

Then
PF + PF' = 2a

=> √[(x - c)² + y²] + √[(x + c)² + y²] = 2a
√[(x + c)² + y²] = 2a - √[(x-c)² + y²]

On squaring both sides,

we get
[(x + c)² + y²] = 4a² + (x - c)² + y² - 4a√[(x - c)² + y²]
[(x + c) ² - (x - c) ²] - 4a² = - 4a √[(x - c)² + y²]
4 x c – 4a² = - 4a √[(x - c)² + y²]
√[(x - c)² + y²] = a – (c/a) x

Again squaring on both sides ,

we get
(x - c)² + y² = a² + [c²x²/a²] - 2cx
x² - [c²x²/a²] + y² = a² - c²
x² [1 – (c²/a²)] + y² = a² - c²
[x²(a² - c²)] / a² +y² = a² - c²

Dividing by (a² - c²)

we get
(x²/a²) + (y²/(a²- c²)) = 1
(x²/a²) + (y²/b²) = 1 , where b² = a² - c²

Thus (x²/a²) + (y²/b²) = 1 is the required equation of an ellipse in standard form

In our next blog we shall learn about relative molecular mass I hope the above explanation was useful.Keep reading and leave your comments.