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Ellipse - Notes, Concept and All Important Formula

ELLIPSE

1. STANDARD EQUATION & DEFINITION :

Standard equation of an ellipse referred to its principal axis along the co-ordinate axis is \(\dfrac{\mathbf{x}^{2}}{\mathbf{a}^{2}}+\dfrac{\mathbf{y}^{2}}{\mathbf{b}^{2}}=\mathbf{1}\). where \(a>b \) & \( b^{2}=a^{2}\left(1-e^{2}\right)\)
\(\Rightarrow a^{2}-b^{2}=a^{2} e^{2} .\)
where \(e=\) eccentricity \((0<e<1)\).
Standard equation and definition of Ellipse
\(\mathrm{FOCI}: \mathrm{S} \equiv(\mathrm{ae}, 0) \) & \( \mathrm{~S}^{\prime} \equiv(-\mathrm{ae}, 0) .\)

(a) Equation of directrices :

\(\mathrm{x}=\dfrac{\mathrm{a}}{\mathrm{e}} \) & \( \mathrm{x}=-\dfrac{\mathrm{a}}{\mathrm{e}} \text { . }\)

(b) Vertices:

\(\mathrm{A}^{\prime} \equiv(-\mathrm{a}, 0) \quad \) & \( \mathrm{~A} \equiv(\mathrm{a}, 0)\)

(c) Major axis : The line segment \(A^{\prime} A\) in which the foci \(S^{\prime}\) & S lie is of length \(2 \mathrm{a} \) & \(\) 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) \(\left(\pm \dfrac{\mathrm{a}}{e}, 0\right)\).

(d) Minor Axis : The y-axis intersects the ellipse in the points \(B^{\prime} \equiv(0,-b) \) & \( B \equiv(0, b)\). The line segment \(B\) 'B of length \(2 \mathrm{~b}(\mathrm{~b}<\mathrm{a})\) is called the Minor Axis of the ellipse.

(e) Principal Axis : The major & minor axis together are called Principal Axis of the ellipse.

(f) Centre : The point which bisects every chord of the conic drawn through it is called the centre of the conic. \(C \equiv(0,0)\) the origin is the centre of the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\)

(g) Diameter: A chord of the conic which passes through the centre is called a diameter of the conic.

(h) Focal Chord : A chord which passes through a focus is called a focal chord.

(i) Double Ordinate : A chord perpendicular to the major axis is called a double ordinate with respect to major axis as diameter.

(j) Latus Rectum : The focal chord perpendicular to the major axis is called the latus rectum.

(i) Length of latus rectum

\(\left(\mathrm{LL}^{\prime}\right)=\dfrac{2 \mathrm{~b}^{2}}{\mathrm{a}}=\dfrac{(\text { minor axis })^{2}}{\text { major axis }}=2 \mathrm{a}\left(1-e^{2}\right)\)

(ii) Equation of latus rectum : \(\mathrm{x}=\pm \mathrm{ae}\).

(iii) Ends of the latus rectum are \(\mathrm{L}\left(\mathrm{ae}, \dfrac{\mathrm{b}^{2}}{\mathrm{a}}\right), \mathrm{L}^{\prime}\left(\mathrm{ae},-\dfrac{\mathrm{b}^{2}}{\mathrm{a}}\right)\),

\(\mathrm{L}_{1}\left(-\mathrm{a} e, \dfrac{\mathrm{b}^{2}}{\mathrm{a}}\right) \text { and } \mathrm{L}_{1} \cdot\left(-\mathrm{ae},-\dfrac{\mathrm{b}^{2}}{\mathrm{a}}\right)\)

(k) Focal radii: \( \mathrm{SP}=\mathrm{a}-e \mathrm{x}\) and \(\mathrm{S}^{\prime} \mathrm{P}=\mathrm{a}+\mathrm{ex}\)

\(\Rightarrow \mathrm{SP}+\mathrm{S}^{\prime} \mathrm{P}=2 \mathrm{a}=\) Major axis.

(l) Eccentricity : \(e=\sqrt{1-\dfrac{b^{2}}{a^{2}}}\)


2. ANOTHER FORM OF ELLIPSE :

Another form of a Ellipse, vertical ellipse

\(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,(a<b)\)

(a) \(A A^{\prime}=\) Minor axis \(=2 a\)

(b) \(\mathrm{BB}^{\prime}=\) Major axis \(=2 \mathrm{~b}\)

(c) \(a^{2}=b^{2}\left(1-e^{2}\right)\)

(d) Latus rectum

\(\mathrm{LL}^{\prime}=\mathrm{L}_{1} \mathrm{~L}_{1}^{\prime}=\dfrac{2 \mathrm{a}^{2}}{\mathrm{~b}} \text { . }\) equation \(y=\pm\) be

(e) Ends of the latus rectum are:

\(\mathrm{L}\left(\dfrac{\mathrm{a}^{2}}{\mathrm{~b}},\mathrm{be}\right),\)\( \mathrm{L}^{\prime}\left(-\dfrac{\mathrm{a}^{2}}{\mathrm{~b}}, \mathrm{be}\right),\) \( \mathrm{L}_{1}\left(\dfrac{\mathrm{a}^{2}}{\mathrm{~b}},-\mathrm{be}\right),\) \( \mathrm{L}_{1} \cdot\left(-\dfrac{\mathrm{a}^{2}}{\mathrm{~b}},-\mathrm{be}\right)\)

(f) Equation of directrix \(y=\pm \dfrac{b}{e}\).

(g) Eccentricity : \(e=\sqrt{1-\dfrac{\mathrm{a}^{2}}{\mathrm{~b}^{2}}}\)


3. GENERAL EQUATION OF AN ELLIPSE :

General equation of Ellipse

Let \((a, b)\) be the focus \(S\), and \(l x+m y+n=\) 0 is the equation of directrix. Let \(\mathrm{P}(\mathrm{x}, \mathrm{y})\) be any point on the ellipse. Then by definition.

\(\Rightarrow \mathrm{SP}=e\) PM (e is the eccentricity)
\(\Rightarrow(x-a)^{2}+(y-b)^{2}=e^{2} \dfrac{(l x+m y+n)^{2}}{\left(l^{2}+m^{2}\right)}\)
\(\Rightarrow\left(l^{2}+m^{2}\right)\left\{(x-a)^{2}+(y-b)^{2}\right\}\)\(=e^{2}\{l x+m y+n\}^{2}\)


4. POSITION OF A POINT W.R.T. AN ELLIPSE:

The point \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) lies outside, inside or on the ellipse according as \(\dfrac{\mathrm{x}_{1}^{2}}{\mathrm{a}^{2}}+\dfrac{\mathrm{y}_{1}^{2}}{\mathrm{~b}^{2}}-1><\) or \(=0\)


5. AUXILIARY CIRCLE/ECCENTRIC ANGLE :

AUXILIARY CIRCLE/ECCENTRIC ANGLE OF ELLIPSE

A circle described on major axis as diameter is called the auxiliary circle. Let \(Q\) be a point on the auxiliary circle \(x^{2}+y^{2}=a^{2}\) such that QP produced is perpendicular to the \(\mathrm{x}\) -axis then \(\mathrm{P} \) & \( \mathrm{Q}\) are called as the CORRESPONDING POINTS on the ellipse & the auxiliary circle respectively. ' \(\theta\) ' is called the ECCENTRIC ANGLE of the point \(\mathrm{P}\) on the ellipse \((0 \leq \theta<2 \pi)\).

Note that \(\dfrac{l(\mathrm{PN})}{l(\mathrm{QN})}=\dfrac{\mathrm{b}}{\mathrm{a}}=\dfrac{\text { Semi minor axis }}{\text { Semi major axis }}\)

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".


6. PARAMATRIC REPRESENTATION:

The equations \(\mathrm{x}=\mathrm{a} \cos \theta \) & \( \mathrm{y}=\mathrm{b} \sin \theta\) together represent the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\)
where \(\theta\) is a parameter (eccentric angle). 

Note that if 
\(\mathrm{P}(\theta) \equiv(\mathrm{a} \cos \theta, \mathrm{b} \sin \theta)\) is on the ellipse then ;
\(Q(\theta) \equiv(a \cos \theta, a \sin \theta)\) is on the auxiliary circle.


7. LINE AND AN ELLIPSE :

The line \(y=m x+c\) meets the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) in two real points, coincident or imaginary according as \(\mathrm{c}^{2}\) is \(<=\) or \(>\mathrm{a}^{2} \mathrm{~m}^{2}+\mathrm{b}^{2}\).
Hence \(y=m x+c\) is tangent to the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) if \(c^{2}=a^{2} m^{2}+b^{2}\). The equation to the chord of the ellipse joining two points with eccentric angles \(\alpha \) & \( \beta\) is given by \(\dfrac{x}{a} \cos \dfrac{\alpha+\beta}{2}+\dfrac{y}{b} \sin \dfrac{\alpha+\beta}{2}=\cos \dfrac{\alpha-\beta}{2}\).


8. TANGENT TO THE ELLIPSE \(\mathbf{\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1}:\) 

(a) Point form :

Equation of tangent to the given ellipse at its point \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) is \(\dfrac{\mathrm{xx}_{1}}{\mathrm{a}^{2}}+\dfrac{\mathrm{y} \mathrm{y}_{1}}{\mathrm{~b}^{2}}=1\)

(b) Slope form:

Equation of tangent to the given ellipse whose slope is 'm', is \(y=m x \pm \sqrt{a^{2} m^{2}+b^{2}}\)
Point of contact are \(\left(\dfrac{\pm a^{2} m}{\sqrt{a^{2} m^{2}+b^{2}}}, \dfrac{\mp b^{2}}{\sqrt{a^{2} m^{2}+b^{2}}}\right)\)

(c) Parametric form:

Equation of tangent to the given ellipse at its point \((a \cos \theta, b \sin \theta)\), is \(\dfrac{x \cos \theta}{a}+\dfrac{y \sin \theta}{b}=1\)


9. NORMAL TO THE ELLIPSE \(\mathbf{\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1}:\) 

(a) Point form : Equation of the normal to the given ellipse at \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) \text { is } \dfrac{\mathrm{a}^{2} \mathrm{x}}{\mathrm{x}_{1}}-\dfrac{\mathrm{b}^{2} \mathrm{y}}{\mathrm{y}_{1}}=\mathrm{a}^{2}-\mathrm{b}^{2}=\mathrm{a}^{2} e^{2}\)

(b) Slope form : Equation of a normal to the given ellipse whose slope is 'm' is \(y=m x \mp \dfrac{\left(a^{2}-b^{2}\right) m}{\sqrt{a^{2}+b^{2} m^{2}}}\).

(c) Parametric form : Equation of the normal to the given ellipse at the \(\operatorname{point}(a \cos \theta, b \sin \theta)\) is  \(ax\sec \theta- by \operatorname{cosec} \theta=\left(a^{2}-b^{2}\right)\)


10. CHORD OF CONTACT :

If \(\mathrm{PA}\) and \(\mathrm{PB}\) be the tangents from point \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) to the ellipse \(\dfrac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\dfrac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1\)
then the equation of the chord of contact \(\mathrm{AB}\) is \(\dfrac{\mathrm{xx}_{1}}{\mathrm{a}^{2}}+\dfrac{\mathrm{y} \mathrm{y}_{1}}{\mathrm{~b}^{2}}=1\) or \(\mathrm{T}=0\) at \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\)


11. PAIR OR TANGENTS :

Pair of tangent on ellipse

If \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) be any point lies outside the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) and a pair of tangents PA, \(\mathrm{PB}\) can be drawn to it from \(\mathrm{P}\). Then the equation of pair of tangents of \(\mathrm{PA}\) and \(\mathrm{PB}\) is \(\mathrm{SS}_{1}=\mathrm{T}^{2}\)

where \(\quad S_{1}=\dfrac{x_{1}^{2}}{a^{2}}+\dfrac{y_{1}^{2}}{b^{2}}-1,\)  \( T=\dfrac{x x_{1}}{a^{2}}+\dfrac{y y_{1}}{b^{2}}-1\)

i.e. \(\left(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-1\right)\left(\dfrac{x_{1}^{2}}{a^{2}}+\dfrac{y_{1}^{2}}{b^{2}}-1\right)\)\(=\left(\dfrac{x x_{1}}{a^{2}}+\dfrac{y y_{1}}{b^{2}}-1\right)^{2}\)


12. 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 \(\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{a}^{2}+\mathrm{b}^{2}\) 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.


13. EQUATION OF CHORD WITH MID POINT \(\mathbf{\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)}\) :

The equation of the chord of the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\), whose mid-point be \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) is \(\mathrm{T}=\mathrm{S}_{1}\)
where \(\mathrm{T}=\dfrac{\mathrm{xx}_{1}}{\mathrm{a}^{2}}+\dfrac{y \mathrm{y}_{1}}{\mathrm{~b}^{2}}-1, \mathrm{~S}_{1}=\dfrac{\mathrm{x}_{1}^{2}}{\mathrm{a}^{2}}+\dfrac{\mathrm{y}_{1}^{2}}{\mathrm{~b}^{2}}-1\)
i.e. \(\left(\dfrac{x x_{1}}{a^{2}}+\dfrac{y y_{1}}{b^{2}}-1\right)=\left(\dfrac{x_{1}^{2}}{a^{2}}+\dfrac{y_{1}^{2}}{b^{2}}-1\right)\)


14. IMPORTANT HIGHLIGHTS for \(\mathbf{\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1}\) :

Important highlights of ellipse

(I) The tangent & normal at a point \(\mathrm{P}\) on the ellipse bisect the external & internal angles between the focal distances of \(\mathrm{P}\). 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.

(II) Point of intersection of the tangents at the point \(\alpha \) & \( \beta\) is \(\left(a \dfrac{\cos \frac{\alpha+\beta}{2}}{\cos \frac{\alpha-\beta}{2}}, b \dfrac{\sin \frac{\alpha+\beta}{2}}{\cos \frac{\alpha-\beta}{2}}\right)\)

(III) If \(\mathrm{A}(\alpha), \mathrm{B}(\beta), \mathrm{C}(\gamma) \) & \( \mathrm{D}(\delta)\) are conormal points then sum of their eccentric angles is odd multiple of \(\pi\). i.e. \(\alpha+\beta+\gamma+\delta=(2 \mathrm{n}+1) \pi\).

(IV) If \(A(\alpha), B(\beta), C(\gamma) \) & \( D(\delta)\) are four concyclic points then sum of their eccentric angles is even multiple of \(\pi\). i.e. \(\alpha+\beta+\gamma+\delta=2 \mathrm{n} \pi\)

(V) The product of the length's of the perpendicular segments from the foci on any tangent to the ellipse is \(b^{2}\) and the feet of these perpendiculars lie on its auxiliary circle.


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  1. AnonymousMarch 30, 2025 at 6:08 PM

    Thanks sir! I'm really really grateful to you. Take my regards from Bangladesh.

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