Ellipse is nothing but a circle, in different scales along X-axis and Y-axis

1. Equation of an ellipse from compression of coordinates
Weknow is equation of a circle with its centre at the origin (0, 0) and with unit radius is 2 2
1
X Y
 
. If we replace X with x/a and Y with y/b where a  b, the curve obtained would no more be a circle
in general , as it would have been only in case when b = a. The graph would appear flattened

2. In y-direction as shown in the figure where PMN is a circle represented by
2 2
2 2
1
X Y
a a
  and
QMN is the graph of the modified eqn.
1
b
y
a
x
2
2
2
2


.
All the y coordinates in the new graph have been reduced in the ratio b/a. The curve shall be
called an ellipse with major axis 2a and minor axis 2b; as may be obtained
by putting y = 0 and x = 0 in its eqn. in turn, to get maximum values of x and y respectively and
denoting them by a and b respectively. Its equation would be
given by
2 2
2 2
1
x y
a b
  …………………………………… eqn(1)
This is done as follows :
Let the eqn. of the circle be written in form
1
a
Y
a
X
2
2
2
2

 …………………….(a)
Transform the coordinates to x and y ,only by compressing Y coordinates by
a factor b/a where b/a < 1;i.e.,x = X, and y = b .Y/a ;

3. OR X = x and Y = a .y/b……………………………………………………….(b)
The curve PMN is now changed to the curve QMN because of the transformation of
coordinates to new coordinates x and y. A relationship between x and y shall be its equation.
Since x and y are related to X and Y as in (b) and X and Y are related to each other as in (a),
we can rewrite (a) as,
1
a
b
y
a
a
x
2
2
2
2
2
2

 , 
1
b
y
a
x
2
2
2
2


which is the only one relationship between x any we sought for, and becomes the eqn. of the
new curve ( called ellipse).
The curve shall be bounded by as seen by analyzing the equation 












2
2
a
x
1
b
y
or













2
2
b
y
1
a
x
, (obtained from eqn.(2).), as and no part of the curve shall lie beyond
x = a, x = - a , y = b and y = - b, for, x and y to have real number values.

4. The curve would be symmetrical about both the axes as its eqn. does not change by
replacing x by – x , or y by – y.
We can also have the x- coordinate compressed and y-coordinate stretched
by prescribing b  a to get a standing ellipse with major axis b in y-direction and minor axis a
in x-direction. Again, we can replace x and y with each other to get the said curve too; or
simply interchange the axes which is safe as long as the curve is symmetrical about the axes.
Still in another way, we can stretch only one of the coordinates, x-coordinate say, of the circle
1
b
y
b
x
2
2
2
2

 to get the ellipse
1
b
y
a
x
2
2
2
2


. So any ellipse can be thought of as a
circle which is stretched in direction of one of the axes only.
If b > a, the circle
1
a
y
a
x
2
2
2
2


can be thought of as expanded in the direction of y-axis
or compressed along the x-axis and a standing ellipse is obtained with its major axis 2b along
the y-axis. This is just as if the axes have been interchanged.