The eccentricity of a parabola is the ratio of the distance of any point on the parabola from the focus to its perpendicular distance from the directrix. Since these two distances are equal, the eccentricity of a parabola is 1.
Formula
The eccentricity of a parabola is given by:
e = (Distance of a point from the focus) / (Perpendicular distance of the point from the directrix)
For a parabola, these two distances are always equal. Therefore:
e = 1
Hence, the eccentricity of a parabola is always 1.
Derivation of Eccentricity of Parabola
Consider any point P on a parabola. Let F be the focus and l be the directrix. Draw PM perpendicular to the directrix, where M is a point on the directrix.
According to the definition of a parabola, the distance of any point P from the focus is equal to its perpendicular distance from the directrix. Therefore,
PF = PM
The eccentricity of a parabola is given by:
e = PF / PM
Since PF = PM,
e = PF / PF = 1
Hence, the eccentricity of a parabola is: e = 1
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