ELLIPSE
1. STANDARD EQUATION & DEFINITION :
\(\Rightarrow a^{2}-b^{2}=a^{2} e^{2} .\)
where \(e=\) eccentricity \((0<e<1)\).
(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 :
\(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,(a<b)\)
(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 :
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.
\(\Rightarrow(x-a)^{2}+(y-b)^{2}=e^{2} \dfrac{(l x+m y+n)^{2}}{\left(l^{2}+m^{2}\right)}\)
\(\Rightarrow\left(l^{2}+m^{2}\right)\left\{(x-a)^{2}+(y-b)^{2}\right\}\)\(=e^{2}\{l x+m y+n\}^{2}\)
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 :
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:
where \(\theta\) is a parameter (eccentric angle).
7. LINE AND AN ELLIPSE :
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:
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 :
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 :
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 \(\mathbf{\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)}\) :
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 \(\mathbf{\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1}\) :
(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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