Derivation of Eccentricity of Hyperbola

Let us see how the above mentioned formula for eccentricity can be derived. We know that, eccentricity of a hyperbola is the ratio of the distances from a point on it to the focus and the directrix. By using this definition, we would derive expression for eccentricity of hyperbola for following equation of the hyperbola,

x²/a² – y²/b² = 1

Let the coordinates for Focii for the hyperbola be F(c,0) and F'(-c,0). As per definition of hyperbola, for a point on hyperbola P(x,y).

PF’ – PF = 2a

By using the formula for distance between two points, we get,

√((x+c)²+y²) – √((x-c)²+y²) = 2a

√((x+c)²+y²) = 2a + √((x-c)²+y²)

On squaring both sides, we get,

(x+c)² + y² = 4a² + (x-c)² + y² + 4a√((x-c)²+y²)

On further simplification and rearrangement of terms, we get,

x²/a² + y²/(c²-a²) = 1

On comparing with the equation of hyperbola, we get,

c² – a² = b²

Now, as per definition of hyperbola,

e = c/a

where,

c = distance of a point on hyperbola from focus,

a = distance of the point from the directrix.

If we take the point on hyperbola, as vertex of the hyperbola. Then, we get after substituting the required distances in above equation,

e = √(a²+b²)/a

or, e = √(1 + b²/a²)

Thus, we have derived the expression for eccentricity of the hyperbola in terms of lengths of its major axis and minor axis.

Eccentricity of Hyperbola refers to the deviation of the conic section from being circular and closeness towards being oval in shape. In other words, it can be defined as the measure of how flattened or elongated a hyperbola is. It is calculated as the ratio of the distance of a point on the hyperbola from its focus and the directrix. This ratio for a hyperbola is always greater than one, which implies that the two branches of the hyperbola diverge away from each other when extended to infinity. It is denoted by the letter ‘e’. It can be used to predict the shape of the hyperbola.

In this article, we will discuss, the eccentricity of a hyperbola, its formula, derivation, solved examples, practice problems and related frequently asked questions.