Parabola

The word parabola originated as a combination of 2 Greek words, para, which means besides and bole, which means throw. When any heavy object is thrown in air, its trajectory is a parabola. Hence the name.

The eccentricity of parabola, {e} is {1}. Let {S(a,0)} be the focus and {d \equiv x+a=0} be the directrix. According to the focus-directrix property,

{SP = e PM \ and \ e =1, \ \therefore \ SP= PM,}

where {P(x,y)} is any point on the parabola. Using this condition for locus of a point, we get the standard equation of parabola,

{y^2 = 4ax, \ a>0}

 

  • It is symmetric about {X} axis, and it extends to infinity to the right of {Y} axis
  • Focal distance = {x_1 + a}
  • Latus rectum = {4a}
  • Parametric equations : {x= at^2, \ y=2at}, where {t} is the parameter
  • The general equation of parabola is of the form {y=ax^2+bx+c} or {x=ay^2+by+c , a \ne 0}. It can be converted to the standard form by shift of origin and (sometimes) rotation of axes.

 

 

  • Where can you see the parabolic shape?

I) Shape of satellite dish

II) Automobile Headlights (The dim-dip feature)

III) McDonald’s Arches

IV) Mirror Furnace (capable of producing temperatures up to {3300^oC} from rays of the Sun)

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