SHORT NOTES: Hyperbola

F,F'=foci
L,L'=directrices
A,A'=centres
Standard equation of Hyperbola:

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

conjugate hyperbola(axis is y axis)

$-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Parametric form:

$x=a\sec \theta, \ y=b\tan \theta$

Tangent to hyperbola:

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

Point form at $\left ( x_1,y_1 \right )$ : $\frac{xx_1}{a^2}-\frac{yy_1}{b^2}=1$

Normal to hyperbola:

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

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

For rectangular hyperbola both transverse and conjugate axes are same So eccentricity e is $\sqrt{2}$ .

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Hyperbola

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