__BASIC CONCEPTS__

An ellipse is the locus of a point which moves so that its distance from a fixed point (**called focus**) bears to its distance from a fixed straight line (**called directrix**) a constant ratio (**called eccentricity**) which is less than unity.

Ellipse is also defined as the locus of a point the sum of whose distances from two fixed points is a constant and greater than the distance between the two fixed points.

**Illustration :** Identify the curve

**Solution:** Since represents distance between points (x, y) and (−1, 0). The given equation states that the point P(x, y) moves in such a way that its sum of distances from points (−1, 0) and (0, 1) is constant i.e. 2 , and which is also greater than distance between two points. Hence the locus is an ellipse with foci (−1 , 0) and (0 , 1).

__Standard Equation:__

Let ZN be the directrix, S the focus and e the eccentricity of the ellipse whose equation is required.

Draw SZ perpendicular to ZN. We can divide ZS both internally and externally in the ratio e : 1

i.e. if the points of divisions be A and A’ , as shown in the figure, then

AS = e.ZA and A’S = e.ZA’

Then by definition of the ellipse A and A’ lie on the ellipse. Let C be the middle point of AA’ and let AA’ = 2a.

Let the x-axis be along AA’ and the y-axis be passing through C.

Then C is the origin and AC = a = CA’.

Also AS = AC − SC = a − SC = e(ZA) = e(ZC − a) and A,S = a + SC = e(ZC + a)

=> 2a = 2e(ZC)

=> ZC = a/e and SC = ae.

Therefore S ≡ (−ae , 0) and equation of the line NZ is x = − a/e

Now any point P(x , y) on the ellipse will satisfy SP^{2} = e^{2} (perpendicular distance of P from ZN)^{2}

=> (x + ae)^{2} + (y − 0)^{2} = e^{2}(x + a/e)^{2}

=> x^{2}(1 − e^{2}) + y^{2} = a^{2}(1 − e^{2})

**Therefore,**

or ,

where b^{2} = a^{2}(1 − e^{2})

The eccentricity of the ellipse is given by the relation

b^{2} = a^{2}(1 − e^{2}), i.e., e^{2} = 1 − b^{2}/a^{2}

Since ellipse is symmetrical about the y-axis , it follows that there exists another focus S’ at (ae, 0) and a corresponding directrix N’Z’ , with the equation x = a/e , such that the same ellipse is described if a point moves so that its distance from S’ is e times its distance from N’Z’ .

Thus, an ellipse has two foci and two directrices.