# Standard Equations of an Ellipse

To obtain an equation of the ellipse in a simplified form, we place the centre of the ellipse at the origin and position the ellipse so that its major axis coincides with one of the coordinate axis, say x-axis. If c is the distance from one focus to the centre, then one focus will be at F

_{1 }= (-c, 0) and the other one at F

_{2 }= (c, 0). We take the constant distance (sum of distances of point P from the foci F

_{1}and F

_{2}) to be 2a. We shall take a>c. If P(x, y) is any point on the ellipse (Figure given above.) then by definition of the ellipse

PF

_{1 }+ PF

_{2 }= 2a

Squaring both the sides, we get

(x + c)

^{2 }+ y

^{2 }= 4a

^{2 }- 4a

Squaring both the sides, we get

^{}

Dividing both the sides by a

^{2}(a

^{2 }- c

^{2}), we get

If we put b

^{2 }= a

^{2 }- c

^{2}where b > 0 then the above equation reads as

**Alternative Method**

To obtain an equation of the ellipse in a simplified form from the alternative definition. We proceed as follows Let F

_{1}be the focus and D

_{1}be the directrix of the ellipse (Figure given below.). Let M

_{1}be the foot of perpendicular from F

_{1}to D

_{1}.

# Divide F1M1 internally and externally at V1 and V2, respectively in the ratio e:1, so that

-----------------(1)

_{1}and V

_{2}lie on the ellipse.

Let O be the mid-point of V

_{1}V

_{2}suppose V

_{1}V

_{2 }= 2a.

Then OV

_{1 }= OV

_{2 }= a.

From (1), we get F

_{1 }V

_{1 }= e V

_{1}M

_{1}and F

_{1}V

_{2 }= e V

_{2}M

_{1 }------------------(2)

Adding, we get F

_{1}V

_{1 }+ F

_{1 }V

_{2 }= e(V

_{1}M

_{1 }+ V

_{2}M

_{1})

â‡’ V

_{1}V

_{2 }= e[(OM

_{1 }- OV

_{1}) + (OM

_{1 }+ OV

_{2})]

â‡’ 2a = 2e(OM

_{1}) [OV

_{1 }= OV

_{2 }= a]

â‡’ OM

_{1 }= .

Subtracting the relations in (2), we get F

_{1}V

_{2}âˆ’ F

_{1}V

_{1 }= e V

_{2}M

_{1 }- e V

_{1}M

_{1}

â‡’ (OF

_{1 }+ OV

_{2}) âˆ’ (OV

_{1 }âˆ’ OF

_{1}) = e(V

_{2}M

_{1 }âˆ’ V

_{1}M

_{1})

â‡’ 2OF

_{1 }= e (V

_{1}V

_{2}) [OV

_{2 }= OV

_{1 }= a]

â‡’ OF

_{1 }= ae

Take O as the origin, OV

_{1}as the x-axis and OY perpendicular to OV

_{1}as y-axis.

Since F

_{1}is to the left of the y-axis and OF

_{1 }= ae, the coordinates of F

_{1}are (âˆ’ ae, 0) and since OM

_{1 }= , the equation of the directrix D

_{1}is .

Let P(x, y) be any point on the ellipse. Let M be the foot of perpendicular from P to D

_{1}. By definition of the ellipse

â‡’ F

_{1}P

^{2 }= e

^{2}PM

^{2}-------------------(3)

We have F

_{1}P

^{2 }= (x + ae)

^{2 }+ y

^{2}and PM = .

Substituting these values in (3), we get

â‡’ x

^{2 }+ 2xae + a

^{2 }e

^{2 }+ y

^{2 }= e

^{2}x

^{2 }+ 2aex + a

^{2}

â‡’ x

^{2 }(1 âˆ’ e

^{2}) + y

^{2 }= a

^{2}(1 âˆ’ e

^{2})

Dividing throughout by a

^{2 }(1 âˆ’ e

^{2}), we get

Since 0 < e < 1, a

^{2 }(1 âˆ’ e

^{2}) > 0. We take a

^{2 }(1 âˆ’ e

^{2}) = b

^{2 }(b > 0), so that the above equation becomes

------------------- (4)

Conversely, every equation of the form (4) represents an ellipse.

If P (x, y) satisfies the equation (4), then by retracing our steps we can show that PF

_{1 }= e PM, so that the point P (x, y) lies on the ellipse. Hence, the equation

represents an ellipse.