Showing posts with label equation of ellipse. Show all posts
Showing posts with label equation of ellipse. Show all posts

Thursday, September 16

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.