Two Dimensional Geometry Ellipse

Know all about Ellipse with Examples

 

Definition of Ellipse

If in a plane, S and L represent a fixed point and a fixed straight line respectively and a moving point P rotates in that plane in such a way that the ratio of its distance from the fixed point S and the fixed straight line L is always constant and the value of this constant is less than 1 then, the locus of the moving point P is called ellipse.

The fixed point S is called the focus and the fixed straight line L is called directirx of the ellipse. The constant ratio is eccentricity of the ellipse and is denoted as e. In the case of ellipse, e < 1.

 

A few Definitions

\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1……….\left( 1 \right)\]

The point where the perpendicular from the focus to the directrix intersects the ellipse is called the vertex of the ellipse. In the above figure, perpendicular from the focus to the directrix intersects the ellipse at A and A’. Therefore ellipse has two vertices A and A’ and their coordinates are (a, 0) and (–a, 0) respectively.

The mid-point of the line joining the two vertices of the ellipse is the center of the ellipse. In the figure C is the mid-point of the line AA’ joining the vertices A and A’, hence the center of the ellipse (1) and its coordinates is (0, 0)

The joining line AA’ of the vertices A and A’ is the major axis of the ellipse. In ellipse (1) x-axis the major axis and its length is 2a units.

Putting x = 0 in (1) we get y=\pm \,b. Therefore the ellipse interests the y-axis at the point B(0, b) and B’(0, –b). The line BB’ is called the minor axis of the ellipse (1). In ellipse (1) y-axis is the minor axis and its length is 2b units.

 

Important Properties of Ellipse

Let P(x, y) be any point on the ellipse (1) and PN and PN’ are the perpendiculars from P to the major axis and minor axis respectively, then

\[\left( i \right)\frac{P{{N}^{2}}}{AN\times A’N}=\frac{{{b}^{2}}}{{{a}^{2}}}\]

\[\left( ii \right)\frac{P{{N}^{2}}}{BN’\times B’N’}=\frac{{{a}^{2}}}{{{b}^{2}}}\]

\[\left( iii \right)\,SP+S’P=2a\]

Where AA’ is major axis, BB’ is minor axis, S and S’ are two focus and 2a, 2b are the lengths of major axis and minor axis respectively.

 

Some Formulae on Ellipse

1. In the following table results of various ellipse are shown:

 \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] \[\left[ {{a}^{2}}>{{b}^{2}} \right]\]\[\frac{{{x}^{2}}}{{{b}^{2}}}+\frac{{{y}^{2}}}{{{a}^{2}}}=1\,\] \[\left[ {{a}^{2}}>{{b}^{2}} \right]\]\[\frac{{{\left( x-\alpha  \right)}^{2}}}{{{a}^{2}}}+\frac{{{\left( y-\beta  \right)}^{2}}}{{{b}^{2}}}=1\] \[\left[ {{a}^{2}}>{{b}^{2}} \right]\]\[\frac{{{\left( x-\alpha  \right)}^{2}}}{{{b}^{2}}}+\frac{{{\left( y-\beta  \right)}^{2}}}{{{a}^{2}}}=1\] \[\left[ {{a}^{2}}>{{b}^{2}} \right]\]
Major axisx-axisy-axisParallel to x-axisParallel to y-axis
Minor axisy-axisx-axisParallel to y-axisParallel to x-axis
Equation of major axisy = 0x = 0y = βx = α
Equation of minor axisx = 0y = 0x = αy = β
Length of major axis2a units2a units2a units2a units
Length of minor axis2b units2b units2b units2b units
Coordinates of center(0, 0)(0, 0)(α, β)(α, β)
Coordinates of two vertex(± a, 0)(0, ± a)(α ± a, β)(α, β ± a)
eccentricity\[e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}\]\[e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}\]\[e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}\]\[e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}\]
Coordinates of two focus(± ae, 0)(0, ± ae)(α ± ae, β)(α, β ± ae)
Distance between two focus2ae units2ae units2ae units2ae units
Length of latus rectum\[\frac{2{{b}^{2}}}{a} units\]\[\frac{2{{b}^{2}}}{a} units\]\[\frac{2{{b}^{2}}}{a} units\]\[\frac{2{{b}^{2}}}{a} units\]
Coordinates of 4  end points of the two latus rectum\[\left( ae,\,\pm \frac{{{b}^{2}}}{a} \right)\] \[\left( -ae,\,\pm \frac{{{b}^{2}}}{a} \right)\]\[\left( \pm \frac{{{b}^{2}}}{a},\,ae \right)\] \[\left( \pm \frac{{{b}^{2}}}{a},\,-ae \right)\]\[\left( \alpha \pm ae,\,\beta \pm \frac{{{b}^{2}}}{a} \right)\]\[\left( \alpha \pm \frac{{{b}^{2}}}{a},\,\beta \pm ae \right)\]
Equation of two latus rectum\[x=\pm \,ae\]\[y=\pm \,ae\]\[x=\alpha \pm \,ae\]\[y=\beta \pm \,ae\]
Equation of two directrix\[x=\pm \,\frac{a}{e}\]\[y=\pm \,\frac{a}{e}\]\[x=\alpha \pm \,\frac{a}{e}\]\[y=\beta \pm \,\frac{a}{e}\]
Distance between two directrix\[\frac{2a}{e} units\]\[\frac{2a}{e} units\]\[\frac{2a}{e} units\]\[\frac{2a}{e} units\]

2. If S and S’ are the two focus and P(x, y) be any point on the ellipse \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\,\,\left[ {{a}^{2}}>{{b}^{2}} \right] then SP = a – ex, S’P = a + ex and SP + S’P = 2a units.

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