# Hyperbola

NOTE: We are going to use the section formula to derive the equation of hyperbola, as we did for ellipse. Since there are 2 variants of the formula (external and internal division), we get 2 foci and 2 directrices.

A hyperbola is a conic section, whose eccentricity ${e}$ is greater than 1. In other words, if ${S(ae,0)}$ is a focus and ${x= \frac {a}{e}}$ is the directrix, then for any point ${P(x,y)}$,

${SP = e PM, \ and \ e>1}$

The standard equation of hyperbola is

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

• Hyperbola is NOT a closed curve, has 2 parts, which are mirror images of each other
• Symmetric about ${X}$ and ${Y}$ axes
• Does not pass through the origin
• Intersection with ${X}$ axis – at ${(a,0)}$ and ${(-a,0)}$
• Does not intersect ${Y}$ axis
• Foci : ${(ae,0)}$ and ${(-ae,0)}$, directrices : ${x \pm \frac a e =0}$
• Difference of focal distances = ${2a}$ = constant (We’ve used this as the condition for locus of a point)
• Length of latus rectum = ${\frac {2b^2}{a}}$
• Parametric Equations : ${x = a sec (\theta), \ y= b tan (\theta)}$
• Transverse axis has length ${2a}$conjugate axis has length ${2b}$
• If ${a=b}$, we get a rectangular hyperbola

• ### Where can you see the hyperbolic shape?

I) The black lines on a basketball or the red lines on the baseball

II) Orbits of comets around the Sun (or any star)

III) Interference patterns by 2 circular waves

IV) Potato chips (:P)

V) Cooling towers in an industrial plant

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