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).
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 SP2 = e2 (perpendicular distance of P from ZN)2
=> (x + ae)2 + (y − 0)2 = e2(x + a/e)2
=> x2(1 − e2) + y2 = a2(1 − e2)
where b2 = a2(1 − e2)
The eccentricity of the ellipse is given by the relation
b2 = a2(1 − e2), i.e., e2 = 1 − b2/a2
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.