Ellipse

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

An ellipse is a conic section, whose eccentricity {e} is less 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 ellipse is

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

  • Ellipse is horizontal, when {a>b} i.e. more stretched in {X} direction, vertical, when {b>a}
  • Closed curve
  • Symmetric about {X} and {Y} axes
  • Does not pass through the origin
  • Intersection with {X} axis – at {(a,0)} and {(-a,0)}
  • Intersection with {Y} axis – at {(0,b)} and {(0,-b)}
  • Foci : {(ae,0)} and {(-ae,0)}, directrices : {x \pm \frac a e =0}
  • Sum 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 cos (\theta), \ y= b sin (\theta)}

 

  • Where can you find the elliptic shape?

I) Earth’s orbit around Sun, with Sun as one of the foci of the ellipse

II) The rugby ball

III) Batman logo (boundary)

IV) Eggs, lemons

V) Whispering galleries

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Posted in XI

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