SHORT NOTES: Ellipse

Standard equation of the ellipse :

${\color{Red} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ , where $b^2=a^2\left ( 1-e^2 \right )$

For other form of the ellipse a > b.

Parametric form:

$x=a\cos \phi \ y=b\sin \phi$

Tangent to ellipse :

Point form at $P({x_1},{y_1})$ : ${\dfrac{{x{x_1}}}{{{a^2}}} + \dfrac{{y{y_1}}}{{{b^2}}} = 1}$

Slope form : $y=mx\pm\sqrt{a^2m^2+b^2}$ at $\left ( \pm \frac{a^2m}{\sqrt{a^2m^2+b^2}},\mp \frac{b^2}{\sqrt{a^2m^2+b^2}} \right )$

Normal :

Point form at $P({x_1},{y_1})$ : $\frac{a^2x}{x_1}-\frac{b^2y}{y_1}=a^2-b^2$

Slope form: $y=mx\pm\frac{m\left ( a^2-b^2 \right )}{\sqrt{a^2+b^2m^2}}$ at $\left ( \pm\frac{a^2}{\sqrt{{a^2+b^2m^2}}},\pm\frac{b^2m}{\sqrt{{a^2+b^2m^2}}} \right )$

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Ellipse

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