Important Formulas: Ellipse | Mathematics (Maths) for JEE Main & Advanced PDF Download

Ellipse

Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is Where a > b & b² = a²(1 − e²)  a² − b²= a² e². Where e = eccentricity (0 < e < 1). FOCI : S ≡ (a e, 0) & S' ≡ (− a e, 0).Equations Of Directrices :

Vertices :
A' ≡ (− a, 0) & A ≡ (a, 0) .
Major Axis :
The line segment A' A in which the foci S' & S lie is of length 2a & is called the major axis (a > b) of the ellipse. Point of intersection of major axis with directrix is called the foot of the directrix (z).
Minor Axis:
The y−axis intersects the ellipse in the points B¢ º (0, − b) & B' (0, b). The line segment B'B of length 2b (b < a) is called the Minor Axis of the ellipse.
Principal Axis :
The major & minor axis together are called Principal Axis of the ellipse.
Centre :
The point which bisects every chord of the conic drawn through it is called the centre of the conic. C ≡ (0,0) the origin is the centre of the ellipse Diameter :
A chord of the conic which passes through the centre is called a diameter of the conic.
Focal Chord:
A chord which passes through a focus is called a focal chord.
Double Ordinate :
A chord perpendicular to the major axis is called a double ordinate.
Latus Rectum :
The focal chord perpendicular to the major axis is called the latus rectum. Length of latus rectum (LL') = (distance from focus to the corresponding directrix).Note :
(i) The sum of the focal distances of any point on the ellipse is equal to the major Axis. Hence distance of focus from the extremity of a minor axis is equal to semi major axis. i.e. BS = CA.
(ii) If the equation of the ellipse is given as & nothing is mentioned, then the rule is to assume that a > b.

Position Of A Point W.R.T. An Ellipse :
The point P(x₁, y₁) lies outside, inside or on the ellipse according as ;3. Auxiliary Circle / Eccentric Angle :
A circle described on major axis as diameter is called the auxiliary circle.
Let Q be a point on the auxiliary circle x² + y² = a² such that QP produced is perpendicular to the x-axis then P & Q are called as the Corresponding on the ellipse (0 ≤ θ < 2π).
Hence “ If from each point of a circle perpendiculars are drawn upon a fixed diameter then the locus of the points dividing these perpendiculars in a given ratio is an ellipse of which the given circle is the auxiliary circle”.

Parametric Representation :
The equations x = a cos θ & y = b sin q together represent the ellipse Where θ is a parameter. Note that if P(θ) ≡ (a cos θ, b sin θ) is on the ellipse then ;Q(θ) ≡ (a cos θ, a sin θ) is on the auxiliary circle.

Line And An Ellipse :
The line y = mx + c meets the ellipse in two points real, coincident or imaginary according as c² is < = or > a²m² + b². Hence y = mx + c is tangent to the ellipse The equation to the chord of the ellipse joining two points with eccentric angles α & β is given by 

Tangents :
(i) is tangent to the ellipse at (x₁, y₁).
Note : The figure formed by the tangents at the extremities of latus rectum is rhoubus of area
(ii) is tangent to the ellipse for all values of m.
Note that there are two tangents to the ellipse having the same m, i.e. there are two tangents parallel to any given direction.
(iii) is tangent to the ellipse at the point (a cos θ, b sin θ).
(iv) The eccentric angles of point of contact of two parallel tangents differ by π. Conversely if the difference between the eccentric angles of two points is p then the tangents at these points are parallel.
(v) Point of intersection of the tangents at the point α & β is a 

Normals :
(i) Equation of the normal at (x₁, y₁) is 

(ii) Equation of the normal at the point (acos θ, bsin θ) is ; ax. sec
θ − by. cosec q = (a²− b²).
(iii) Equation of a normal in terms of its slope 'm' is y = mx − 

Director Circle :
Locus of the point of intersection of the tangents which meet at right angles is called the Director Circle. The equation to this locus is x² + y² = a² + b² i.e. a circle whose centre is the centre of the ellipse & whose radius is the length of the line joining the ends of the major & minor axis.

Chord of contact, pair of tangents, chord with a given middle point, pole & polar are to be interpreted as they are in parabola.

Diameter :
The locus of the middle points of a system of parallel chords with slope 'm' of an ellipse is a straight line passing through the centre of the ellipse, called its diameter and has the equation y = 

Important Highlights :
Refering to an ellipse H − 1 If P be any point on the ellipse with S & S¢ as its foci then l (SP) + l (S'P) = 2a.
H − 2 The product of the length’s of the perpendicular segments from the foci on any tangent to the ellipse is b² and the feet of these perpendiculars Y,Y' lie on its auxiliary circle.The tangents at these feet to the auxiliary circle meet on the ordinate of P and that the locus of their point of intersection is a similiar ellipse as that of the original one. Also the lines joining centre to the feet of the perpendicular Y and focus to the point of contact of tangent are parallel.
H − 3 If the normal at any point P on the ellipse with centre C meet the major & minor axes in G & g respectively, & if CF be perpendicular upon this normal, then
(i) PF. PG = b²
(ii) PF. Pg = a²
(iii) PG. Pg = SP. S' P
(iv) CG. CT = CS²
(v) locus of the mid point of Gg is another ellipse having the same eccentricity as that of the original ellipse.
[where S and S' are the focii of the ellipse and T is the point where tangent at P meet the major axis]
H − 4 The tangent & normal at a point P on the ellipse bisect the external & internal angles between the focal distances of P. This refers to the well known reflection property of the ellipse which states that rays from one focus are reflected through other focus & vice−versa. Hence we can deduce that the straight lines joining each focus to the foot of the perpendicular from the other focus upon the tangent at any point P meet on the normal PG and bisects it where G is the point where normal at P meets the major axis.
H − 5 The portion of the tangent to an ellipse between the point of contact & the directrix subtends a right angle at the corresponding focus.
H − 6 The circle on any focal distance as diameter touches the auxiliary circle.
H − 7 Perpendiculars from the centre upon all chords which join the ends of any perpendicular diameters of the ellipse are of constant length.
H − 8 If the tangent at the point P of a standard ellipse meets the axis in T and t and CY is the perpendicular on it from the centre then,
(i) T t. PY = a² − b² and
(ii) least value of Tt is a + b.