Table of Contents

**Definition of Ellipse**

If in a plane, ** S** and

**represent a fixed point and a fixed straight line respectively and a moving point**

*L***rotates in that plane in such a way that the ratio of its distance from the fixed point**

*P***and the fixed straight line**

*S***is always constant and the value of this constant is less than 1 then, the locus of the moving point**

*L***is called ellipse.**

*P*The fixed point ** S** is called the

**focus**and the fixed straight line

**is called**

*L***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

**. Therefore ellipse has two vertices**

*A’***and**

*A***and their coordinates are (a, 0) and (–a, 0) respectively.**

*A’*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

**joining the vertices**

*AA’***and**

*A***, hence the center of the ellipse (1) and its coordinates is (0, 0)**

*A’*The joining line ** AA’** of the vertices

**and**

*A***is the**

*A’***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

**(0, –b). The line BB’ is called the**

*B’***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 axis | x-axis | y-axis | Parallel to x-axis | Parallel to y-axis |

Minor axis | y-axis | x-axis | Parallel to y-axis | Parallel to x-axis |

Equation of major axis | y = 0 | x = 0 | y = β | x = α |

Equation of minor axis | x = 0 | y = 0 | x = α | y = β |

Length of major axis | 2a units | 2a units | 2a units | 2a units |

Length of minor axis | 2b units | 2b units | 2b units | 2b 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 focus | 2ae units | 2ae units | 2ae units | 2ae 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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