Ellipse : Basic Concepts , Standard Equation of ellipse , Position of a Point Relative to an Ellipse

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 :

$\large \sqrt{(x+1)^2 + y^2} + \sqrt{x^2 + (y-1)^2} = 2$

Solution: Since $\large \sqrt{(x+1)^2 + y^2}$ 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 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)

Therefore,

$\large \frac{x^2}{a^2} + \frac{y^2}{a^2(1-e^2)} = 1$

or , $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

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.

Definitions Associated with Ellipse:
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 $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is (0, 0)

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:

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 = b2 / a

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

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 $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , a2 < b2, 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

$\large e = \sqrt{1-\frac{a^2}{b^2}}$

The length of the semi-latus rectum becomes a2/ 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

$\large \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

(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

PS2 = e2 PM2

=> (x − h)2 + (y − k)2

$\large e^2 = \frac{(ax+by+c)^2}{a^2+b^2}$

which is of the form

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 , … (*) where

Δ = abc + 2 fgh − af2 − bg2 − ch2 ≠ 0, h2 < 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:

The point P(x1, y1) is outside or inside or on the ellipse $\large \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ according as the quantity

S1 ≡ $\large (\frac{x^2}{a^2} + \frac{y^2}{b^2} – 1)$

is positive or negative or zero.

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