The general second degree equation in two unknowns can be represent in the form

§2. Second order curves.

The general second degree equation in two unknowns can be represent in the form

and describes the second order curves.

For simplifying an equation, some time we need to move the coordinate axes to a new position.

1) We displaced axes and to the positions and such way, that and . The origin displaced to the point , then

and .

2) We rotated axes about origin in the counterclockwise direction through an angel , then

and .

It is always possible to find a rotation that produces an equation in which no cross- product term

.

Remark. The coefficients and free term in equation are different from its values in equation .

Lets to complete to the squares

.

Let us to make parallel translation with new origin .

We got an equation

, ,

where .

This equation can determine different locus on the plane:

1. – the line of elliptical type;

2. – the line of hyperbolic type;

3. – the line of parabolic type.

1. The case and gives

– equation of the circle.

and gives

– equation of the ellipse.

– major axis;

– minor axis;

– semi-axis respectively;

– center-to-focus distance;

– foci;

– vertices;

– center;

– eccentricity, which describes the various shapes of the ellipse.

2. The case gives

– equation of the hyperbola.

– transverse axis;

– conjugate axis;

– semi-axis respectively;

– center-to-focus distance;

– foci;

– vertices;

– center;

– eccentricity, which describes the various shapes of ellipse;

– equations of asymptotes, which are the lines whose distance to a curve tends to zero, when .

1. The case gives

– equations of the parabolas symmetrical to the axis .

- the equations of the directrix.

Or – equations of the parabolas symmetrical to the axis .

- the equations of the directrix.

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