A hyperbola is a type of conic section formed by the locus of a moving point P, such that the ratio (eccentricity, e) of its distance to a fixed focal point F, to a fixed imaginary line called the directrix D is a positive constant that is greater than 1.
Like the ellipse, the hyperbola is bifocal and therefore also has 2 directrices.
In this video, we formulate the standard equation of the hyperbola with the help of 2 geometric properties:
- We show that the difference between the distance of a point P on the hyperbola to its nearest focus F and from P to the further focus F' is constant. (i.e. PF' - PF = constant)
- By the triangle inequality, the distance from P to the further point (F') is always less than the sum of the distance between the focal points and from P to the closer focal point F. (i.e. PF' < PF + FF')
From this, we can then use the distance formula to derive the standard equation of the hyperbola, which is:
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
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