It is such a locus of a point that moves in such a way that the ratio of its distance from a fixed point called focus and a fixed-line called directrix is constant, which is less than one. (Eccentricity )
Standard Equation of Ellipse:
Standard equation of ellipse to its principal axes along the co-ordinate axes is , where and
Focii : and
Equation of directrices : and
The -axis intersects the ellipse in the points and .
The line segment is of length is called the minor axis of the ellipse.
The line segment in which the focii and lie is of length and is called the major axis of the Ellipse. Point of intersection of major axis with directrix is called the foot of the directrix.
The major and minor axis together is called the principal axis of the ellipse.
Point of intersection of ellipse with the major axis and
A chord that passes through a focus is called a focal chord.
A chord perpendicular to the major axis is called a double ordinate.
The focal chord perpendicular to the major axis is called the latus rectum.
Length of L.R
(distance from focus to the corresponding directrix)
The point which bisects every chord of the conic drawn through it, is called the centre of the conic , is the origin of the centre of the ellipse
Auxiliary Circle/ Eccentric Angle of Ellipse:
A circle described on the major axis of an ellipse as the diameter is called the auxiliary circle.
Hence Eccentric angle of point on the ellipse.
Parametric Representation of Ellipse:
The equation and together represent the ellipse where is a parameter.
Note that if is on the ellipse then; is on the auxiliary circle.
The equation of chord of the ellipse joining two points with eccentric angles and is given by
Equation of Ellipse joining two points and .
Equation of chord is
Position of a Point w.r.t an Ellipse:
The point lies outside, inside or on the ellipse according as or where
Check whether the point lies according to the ellipse
lies inside the ellipse.
Line and an Ellipse:
The line meets the ellipse in two points real, coincident or imaginary according as or
Tangent to Ellipse
(i) Slope form : is tangent to the ellipse , for all values of .
(ii) Point form : is tangent to ellipse at
(iii) Parametric form: is tangent to the ellipse at the point
There are two tangents to the ellipse having the same , i.e there are two tangents parallel to any given direction. These tangents touch the ellipse at extremities of a diameter.
Point of intersection of the tangents at the point and is
The eccentric angles of the points of contact of two parallel tangents differ by .
Normal to Ellipse:
(i) Equation of the normal at to the ellipse is
(ii) Equation of the normal at the point to the ellipse is
(iii) Equation of a normal in terms of its slope is
Some Important Points:
(i) If be any point on the ellipse with focii and , then
(ii) The tangent & normal at a point on the ellipse bisect the external & internal angles between the focal distances of this point.
(iii) The portion of the tangent to an ellipse between the point of contact & the directrix subtends a right angle at the corresponding focus.
(iv) The circle on any focal distance as diameter touches the auxiliary circle. Perpendiculars from the centre upon all the chords which join the ends of any perpendicular diameters of the ellipse are of constant length.
Application of Ellipse:
A whispering gallery is an elliptical-shaped room with a dome-shaped ceiling. If two people stand at the foci of the ellipse and whisper, they can hear each other, but others in this room cannot. “Statuary Hall in the U.S. Capital Building is a whispering gallery.”
Comparative Study of Two Ellipses
Major Axis ,
|7||Relation between and|