Ellipse - Practice Questions & MCQ

Ellipse

An ellipse is the set of all points (x, y) in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).

OR

The locus of a point which moves in a plane such that the ratio of the distance from a fixed point (focus) to the distance from a fixed line (directrix) is constant. Constant is known as eccentricity e and for ellipse e < 1.

Consider the figure, C is the origin, S is the focus and ZM is the directrix.

SA = e.AZ                      (i)

SA’ = e.A’Z                     (ii)

(from the definition)

Adding these 

Let AA’ = 2a. C is the mid point of AA’. Hence, CA = CA’ = a.

A = (a, 0) and A’ = (-a, 0)

From equation (i) and (ii)

SA + SA’ = e(AZ + A’Z)

2a = e(CZ - CA + CA’ + CZ)

2a = e(2CZ)                   [CA = CA’]

CZ = a/e

The equation of directrix, ZM is x = a/e 

Again from equation (i) and (ii)

SA’ - SA = e(A’Z -AZ)

[(CA’ + CS) - (CA -CS)] = e[AA’]

2CS = e(2a)

CS  = ae

The coordinate of focus, S = (ae, 0) and S’ = (-ae, 0) 

P(x, y) is any point on the ellipse and PM is perpendicular to directrix ZM.

Important Terms related to ellipse:

1. Centre: All chord passing through point C is bisected at point C. Here C is the origin, i.e. (0, 0). 

2. Foci: Point S and S’ is foci of the ellipse where, S is (ae, 0) and S’ is (-ae, 0).

3. Directrices: The straight-line ZM and Z’M’ are two directrices of the ellipse and their equations are x = a/e and x = -a/e.

4. Axis: In figure AA’ is called the major axis and BB’ is called the minor axis. 2a is called the length of the major axis and 2b is called the length of the minor axis. 

5. Double Ordinate: If a line perpendicular to the major axis meets the curve at P and P’, then PP’ is called double ordinate. 

6. Latusrectum: Double ordinate passing through focus is called latus rectum. Here LL’ is a latus rectum. There is another latus rectum that passes through the other focus S'. So an ellipse has 2 latus rectum

Standard Equation of Ellipse:

The standard form of the equation of an ellipse with center (0, 0) and major axis on the x-axis is

1. a > b 

2.  the length of the major axis is 2a 

3.  the length of the minor axis is 2b 

4. the coordinates of the vertices are (±a, 0)