# 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)