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Parabola
Definition. The locus of points for which the distance to a fixed point equals the distance to a given straight line (a directrix) is called a parabola.
r
0
To derive the equation of the parabola, we take an arbitrary point Ì(õ;ó) on it and write down the characteristic feature of a parabola as a mathematical formula. The distance from the focus to the directrex is called the parameter of the parabola and denoted by p. Let us find the distance from the point Ì(õ;ó) to the focus:
and By definition, these distances are equal:
Let us transform this, relation by squaring both sides:
We obtain
This is the classical equation of a parabola. The parabola passes through the origin (0;0), because it satisfies equation (20). Suppose that the parameter is a positive number ð>0; then, since ó2>0, we have x>0, and the parabola is contained in the right half-plane. If p<0, then x<0, and the parabola is contained in the left half-plane
p>0 p<0 0 x 0 õ M0(x0,y0)
Consider the equation of a parabola in the “school” form
0 x
0 x
The eccentricity of the parabola, that is, the ratio of the focal radius to the distance from a point to the directrix, equals 1, i.e.,
Date: 2015-01-02; view: 1292 |