Conics

Consider a fixed point S and a fixed line. A conic is the locus of a point P which moves in such a way that the ratio e = is a constant , where PM is the perpendicular distance of P from the fixed line and SP is the distance of P from the point S.

S is called the focus of the conic. The fixed line is called the directrix of the conic and e is called the eccentricity of the conic.

If e = 1, the conic is called a parabola.

If e < 1, the conic is called a ellipse.

If e > 1, the conic is called a hyperbola.

Parabola:

is a parabola with focus at ( a , 0 ) and directrix ' x + a = 0 ' . The vertex of this parabola is at the origin. The x-axis is the axis of symmetry.

See pictures of some parabolas here

Ellipse:

If a > b > 0, then is an ellipse with foci at ( ae , 0 ) and ( - ae , 0 ) and has two directrices and

See pictures of some ellipses here

Hyperbola:

represents a hyperbola with foci at ( ae , 0 ) and ( - ae , 0 ) and has two directrices at and

See pictures of some hyperbolas here

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