Class 11: Ellipse – Lecture Notes

$\displaystyle \textbf{1. Definition of an Ellipse}$
$\displaystyle \text{An ellipse is the locus of a point in a plane which moves in such a way that the ratio of its}$
$\displaystyle \text{distance from a fixed point (called focus) to its distance from a fixed straight line (called}$
$\displaystyle \text{directrix) is always constant and less than unity.}$
$\displaystyle \text{The constant ratio, generally denoted by }e\text{, is called the eccentricity of the ellipse.}$
$\displaystyle \text{If }S\text{ is the focus, }ZZ'\text{ is the directrix and }P\text{ is any point on the ellipse such that}$
$\displaystyle \text{M is the foot of the perpendicular from P on }ZZ',\text{ then}$
$\displaystyle SP=e\cdot PM$
$\displaystyle \text{The equation }ax^2+2hxy+by^2+2gx+2fy+c=0\text{ represents an ellipse if}$
$\displaystyle \Delta=abc+2fgh-af^2-bg^2-ch^2\neq0\quad\text{and}\quad h^2<ab$
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$\displaystyle \textbf{2. Standard Equation of an Ellipse}$
$\displaystyle \text{The equation of the ellipse whose axes are parallel to the coordinate axes and whose centre}$
$\displaystyle \text{is at the origin is}$
$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
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$\displaystyle \textbf{3. Ellipse with Major Axis Along the x-axis }(a>b)$
$\displaystyle \text{Equation: }\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$\displaystyle \text{Centre: }(0,0)$
$\displaystyle \text{Vertices: }(a,0)\text{ and }(-a,0)$
$\displaystyle \text{Foci: }(ae,0)\text{ and }(-ae,0)$
$\displaystyle \text{Length of major axis: }2a$
$\displaystyle \text{Length of minor axis: }2b$
$\displaystyle \text{Major axis: }y=0$
$\displaystyle \text{Minor axis: }x=0$
$\displaystyle \text{Directrices: }x=\frac{a}{e}\text{ and }x=-\frac{a}{e}$
$\displaystyle \text{Eccentricity: }e=\sqrt{1-\frac{b^2}{a^2}}$
$\displaystyle \text{Length of latus-rectum: }\frac{2b^2}{a}$
$\displaystyle \text{Focal distances of a point }(x,y):\;a\pm ex$
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$\displaystyle \textbf{4. Ellipse with Major Axis Along the y-axis }(a<b)$
$\displaystyle \text{Equation: }\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$\displaystyle \text{Centre: }(0,0)$
$\displaystyle \text{Vertices: }(0,b)\text{ and }(0,-b)$
$\displaystyle \text{Foci: }(0,be)\text{ and }(0,-be)$
$\displaystyle \text{Length of major axis: }2b$
$\displaystyle \text{Length of minor axis: }2a$
$\displaystyle \text{Major axis: }x=0$
$\displaystyle \text{Minor axis: }y=0$
$\displaystyle \text{Directrices: }y=\frac{b}{e}\text{ and }y=-\frac{b}{e}$
$\displaystyle \text{Eccentricity: }e=\sqrt{1-\frac{a^2}{b^2}}$
$\displaystyle \text{Length of latus-rectum: }\frac{2a^2}{b}$
$\displaystyle \text{Focal distances of a point }(x,y):\;b\pm ey$
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