Systems Of Conic Sections

The only difference between this lesson and the last one is the curves covered here are not centered on the origin.

Systems of conic sections. In mathematics a conic section or simply conic is a curve obtained as the intersection of the surface of a cone with a plane the three types of conic section are the hyperbola the parabola and the ellipse. Conic sections and standard forms of equations a conic section is the intersection of a plane and a double right circular cone. Conic sections are mathematically defined as the curves formed by the locus of a point which moves a plant such that its distance from a fixed point is always in a constant ratio to its perpendicular distance from the fixed line.

By changing the angle and location of the intersection we can produce different types of conics. Circles ellipses hyperbolas and parabolas. The three types of curves sections are ellipse parabola and hyperbola.

In algebraic geometry the conic sections in the projective plane form a linear system of dimension five as one sees by counting the constants in the degree two equations. Solve systems of quadratic equations. A conic section can be graphed on a coordinate plane.

Parabolas have one focus. The condition to pass through a given point p imposes a single linear condition so that conics c through p form a linear system of dimension 4. You may want to review the concept of completing the square lessons 16 and 17.

This section deals with more ellipses and hyperbolas. None of the intersections will pass through the vertices of the cone. There are four basic types.

The circle is a special case of the ellipse though historically it was sometimes called a fourth type.