Basic Terms Related to an Ellipse :
Let the equation of the ellipse be `x^2/a^2+y^2/b^2=1` .....................(1)
Centre :
In the figure, `C` is the centre of the ellipse. All chords passing through `C` are called diameter and bisected at `C`.
Foci :
`S` and `S'` are the two foci of the ellipse and their coordinates are `( a e, 0)` and `( - a e, 0)` respecitvely.
The line containing two foci are called the focal axis and the distance between `S` & `S'` the focal length.
Directrices :
`ZN` and `Z'N'` are the two direcrices of the ellipse and their equations are `x =a/e` and `x =(-a)/e` respectively.
Here `Z` and `Z'` are called foot of directrix.
Axes :
The Iine segments `A'A` and `B'B` are called the major and minor axes respectively of the ellipse.
The point of intersection of major and minor axis is called centre of the ellipse. Major and minor axis together are called principal axis of ellipse.
Here Semi-major axis are `CA = CA' = a`
and Semi-minor axis are `CB = CB' = b`
Vertex :
The points where major axis meet the ellipse is called its vertices. In the given figure, `A'` and `A` are the vertices of the ellipse.
Ordinate and Double Ordinates :
Let `P` be a point on the ellipse. From `P` we draw `PM` perpendicular to major axis of the ellipse. Produce `PM` to meet the ellipse at `P'`, then `PM` is called an ordinate and `PMP'` is called the double ordinate of the point `P`.
It is also defined as any chord perpendicular to major axis is called its double ordinate.
Latus Rectum :
When double ordinate passes through focus then it is called the Latus rectum.
Let `L'L = 2k`, then `LS = k` so `L = (a e, k)`.
Here `L L'` and `M M'` are called latus rectum.
Since `L(a e, k)` lies on the ellipse `(1)`, therefore `(a^2e^2)/a^2+k^2/b^2 = 1` or `k^2/b^2= 1 - e^2`
or `k^2=b^2(1-e^2)=b^2 . b^2/a^2=b^2/a^2`
`:. k=b^2/a`
Hence length of semi latus rectum `LS = b^2/a=MS'`
i.e. length of the latus rectum `L L'` or `M M'=(2b^2)/a= (text(minor axis)^2)/(text(major axis))`
`=2a(1-e^2)`
`=2e ` (distance from focus to the corresponding directrix).
And the end points of latus rectum are `L(ae,b^2/a),L'(ae,(-b^2)/a), M(-ae,b^2/a)` & `(-ae, -b^2/a)`
Focal Chord :
A chord of the ellipse passing through its focus is called a focal chord.
Focal Distance of a Point :
Let `P(x, y)` be any point on the ellipse
`x^2/a^2+y^2/b^2=1` .................(1)
Then by definition of ellipse,
`SP=ePM=e(MT-PT)=e(2/e-x)=a-ex`
& `S'P=ePM'=e(M'T+PT)=e(a/e+x)=a+ex`
Hence `SP+S'P=2a`
Because of the above property, ellipse is also defined as the locus of a point which moves in a plane such that the sum of its distance from two fixed points (called foci) is a contant (Length of major axis).
This defmtion is called the physical definiition of the ellipse.
Hence `PS + PS' = QS + QS' = TS + TS' =` length of major axis
Centre :
In the figure, `C` is the centre of the ellipse. All chords passing through `C` are called diameter and bisected at `C`.
Foci :
`S` and `S'` are the two foci of the ellipse and their coordinates are `( a e, 0)` and `( - a e, 0)` respecitvely.
The line containing two foci are called the focal axis and the distance between `S` & `S'` the focal length.
Directrices :
`ZN` and `Z'N'` are the two direcrices of the ellipse and their equations are `x =a/e` and `x =(-a)/e` respectively.
Here `Z` and `Z'` are called foot of directrix.
Axes :
The Iine segments `A'A` and `B'B` are called the major and minor axes respectively of the ellipse.
The point of intersection of major and minor axis is called centre of the ellipse. Major and minor axis together are called principal axis of ellipse.
Here Semi-major axis are `CA = CA' = a`
and Semi-minor axis are `CB = CB' = b`
Vertex :
The points where major axis meet the ellipse is called its vertices. In the given figure, `A'` and `A` are the vertices of the ellipse.
Ordinate and Double Ordinates :
Let `P` be a point on the ellipse. From `P` we draw `PM` perpendicular to major axis of the ellipse. Produce `PM` to meet the ellipse at `P'`, then `PM` is called an ordinate and `PMP'` is called the double ordinate of the point `P`.
It is also defined as any chord perpendicular to major axis is called its double ordinate.
Latus Rectum :
When double ordinate passes through focus then it is called the Latus rectum.
Let `L'L = 2k`, then `LS = k` so `L = (a e, k)`.
Here `L L'` and `M M'` are called latus rectum.
Since `L(a e, k)` lies on the ellipse `(1)`, therefore `(a^2e^2)/a^2+k^2/b^2 = 1` or `k^2/b^2= 1 - e^2`
or `k^2=b^2(1-e^2)=b^2 . b^2/a^2=b^2/a^2`
`:. k=b^2/a`
Hence length of semi latus rectum `LS = b^2/a=MS'`
i.e. length of the latus rectum `L L'` or `M M'=(2b^2)/a= (text(minor axis)^2)/(text(major axis))`
`=2a(1-e^2)`
`=2e ` (distance from focus to the corresponding directrix).
And the end points of latus rectum are `L(ae,b^2/a),L'(ae,(-b^2)/a), M(-ae,b^2/a)` & `(-ae, -b^2/a)`
Focal Chord :
A chord of the ellipse passing through its focus is called a focal chord.
Focal Distance of a Point :
Let `P(x, y)` be any point on the ellipse
`x^2/a^2+y^2/b^2=1` .................(1)
Then by definition of ellipse,
`SP=ePM=e(MT-PT)=e(2/e-x)=a-ex`
& `S'P=ePM'=e(M'T+PT)=e(a/e+x)=a+ex`
Hence `SP+S'P=2a`
Because of the above property, ellipse is also defined as the locus of a point which moves in a plane such that the sum of its distance from two fixed points (called foci) is a contant (Length of major axis).
This defmtion is called the physical definiition of the ellipse.
Hence `PS + PS' = QS + QS' = TS + TS' =` length of major axis
