__BASIC CONCEPTS__

__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:__

__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.**

**Definitions Associated with Ellipse**

__Central Curve:__

__Central Curve:__**A curve is said to be a central curve if there is a point, called the centre, such that every chord passing through it is bisected at it.**

**Centre of a central curve is obtained by solving ∂u/∂x = 0 and ∂u/∂y = 0, u = 0 is the equation of the curve and ∂/∂x denotes partial derivative with respect to x ( i.e. treating y as a constant).**

**Obviously ellipse is a central curve and the centre of the ellipse is (0, 0)**

__Diameter:__

__Diameter:__**A chord through the centre of a central curve is called diameter . Diameter can also be defined as the locus of the middle points of a system of parallel chords.**

**The diameter through the foci of the ellipse is called the major axis and the diameter bisecting it at right angle is called the minor axis of the ellipse.**

__Latus Rectum:__

__Latus Rectum:__**The chord through a focus at right angle to the major axis is called the latus rectum.**

**Notes:**

**∎ The major axis AA’ is of length 2a and the minor axis BB’ is of length 2b.**

**∎ The foci are (−ae, 0) and (ae, 0)**

**∎ The equations of the directrices are x = a/e and x = −a/e.**

**∎ The length of the semi latus rectum = b ^{2} / a**

**∎ Circle is a particular case of an ellipse with e = 0**

__Focal Distance of a Point:__

__Focal Distance of a Point:__**Let P(x , y) be a point on the ellipse. Then**

**S’P = ePN’ = e(a/e − x) = a − ex**

**SP = ePN = e ( a/e + x) = a + ex**

**S’P + SP = 2a**

**=> the sum of the focal distances of any point on the ellipse is equal to its major axis.**

**Other forms of Ellipse**

**(i) If in the equation , a ^{2} < b^{2}, then the major and minor axis of the ellipse lie along the y and the x-axes and are of lengths 2b and 2a respectively.**

**The foci become (0, ± be) , and the directrices become**

**y = ± b/e where**

**The length of the semi-latus rectum becomes a ^{2}/ b**

**(ii) If the centre of the ellipse be taken (h, k) and axes parallel to x and y−axes , then the equation of the ellipse is**

**(iii) Let the equation of the directrix of an ellipse be**

**ax + by + c = 0 and the focus be (h , k).**

**Let the eccentricity of the ellipse be e( e < 1 )**

**If P(x , y) is any point on the ellipse, then**

**PS ^{2} = e^{2} PM^{2}**

**=> (x − h) ^{2} + (y − k)^{2}**

**which is of the form**

**ax ^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 , … (*) where**

**Δ = abc + 2 fgh − af ^{2} − bg^{2} − ch^{2} ≠ 0, h^{2} < ab,**

**which are the necessary and sufficient conditions for a general quadratic equation given by (*) to represent an ellipse.**

__Position of a Point Relative to an Ellipse:__

__Position of a Point Relative to an Ellipse:__**The point P(x _{1}, y_{1}) is outside or inside or on the ellipse according as the quantity**

**is positive or negative or zero.**