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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Posted in XI

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