Introduction to Conics

A conic is the intersection of a plane and a right circular cone. When the plane does intersect the vertex of the cone, the resulting conic is called degenerate conic. This include a point, a line and two intersecting lines. Apollonius was the first person to describe conic section as the intersection of a plane with a cone. The equation of every conic can be written in the form AX²+BXY+CX²+DX+EY+F=0. This is the algebraic equation of a conic. .

The equation of every conic can be written in the form AX²+BXY+CX²+DX+EY+F=0. This is the algebraic equation of a conic.

Types of Conics

According to the coefficients of the above equation, this can be classified into circles, parabola, ellipse and hyperbola.

B²- 4AC is called the determinant of the equation. Assuming that the conic is not degenerate, it can be classified.

If B²- 4AC > 0, then the conic is a hyperbola.

If B²- 4AC < 0, then it can be a circle or an ellipse.

If B²- 4AC = 0, then it is a parabola.

Another classification

It can also be classified based on the product AC. Assuming that the conic is not degenerate, it can be classified as follows.

If AC < 0, then it can be an ellipse or a circle.

If AC > 0, then it is a hyperbola.

If AC = 0, and A and C are not both zero, then it is a parabola.

Finally if A= C, then it is a circle.

Common properties of conics

A straight line joining two points on a conic is called a chord, if the chord passes through the focal point it is called a focal chord.

The midpoints of parallel chords lie in a straight line called a diameter. A perpendicular drawn from a point on the axis is called an ordiate and if produced to meet the conic again it is called a double ordinate.

The double ordinate through the focus is called the latus rectum.The auxiliary circle is the circle with its centre on the axis, which contains two vertices.