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 $(0<e<1)$.

$\mathrm{FOCI}: \mathrm{S} \equiv(\mathrm{ae}, 0)$ & $\mathrm{~S}^{\prime} \equiv(-\mathrm{ae}, 0) .$

(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 : $e=\sqrt{1-\dfrac{b^{2}}{a^{2}}}$

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All Chapter Notes, Concept and Important Formula

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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 : $e=\sqrt{1-\dfrac{\mathrm{a}^{2}}{\mathrm{~b}^{2}}}$

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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.

$\Rightarrow \mathrm{SP}=e$ 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}$

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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 $=0$

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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".

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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.

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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 $\dfrac{x}{a} \cos \dfrac{\alpha+\beta}{2}+\dfrac{y}{b} \sin \dfrac{\alpha+\beta}{2}=\cos \dfrac{\alpha-\beta}{2}$.

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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 $\dfrac{x \cos \theta}{a}+\dfrac{y \sin \theta}{b}=1$

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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  $ax\sec \theta- by \operatorname{cosec} \theta=\left(a^{2}-b^{2}\right)$

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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 $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$

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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}$

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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.

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

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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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