A conic section, as the name implies, is a section cut off in various ways from a circular (though not necessarily a right circular) cone by a plane. The shape of the section depends on the position of the cutting plane, that is, whether the plane cuts the cone at particular angles.

Before we proceed, it is best to recapitulate the equation of a line and the condition needed for two lines to be perpendicular.

A linear equation denotes a line. So, ax + by + c = 0 is the equation for a straight line.

by = -ax - c

y = -a/b x - c

This is of the form y = mx + c, where m = -a/b and m = slope of the line.

Suppose we have two lines

Locus

When a point moves in a plane in a manner that always satisfies a given condition, or conditions, the path it traces out is called a locus under these conditions.

Example 1:

The locus of a point that moves at a fixed distance from a fixed point is a circle.

O is the fixed point, OP is the fixed distance or radius and the point P is the locus.

Example 2:

The locus of a point equidistant from two given points is the perpendicular bisector of the line joining the two given points.

Returning to the cone, look at the following diagram.

The point V is called the vertex of the cone. O is the center of the circle formed by the base of the cone. VO is called the axis of the cone and ∠ VOA is called the semi-vertical angle.

Consider a double right circular cone of semi-vertical angle (alpha), and let it be cut by a plane inclined at an angle (theta) to the axis of the cone. We will get different sections (curves) as follows:

Case I

If the plane passes through the vertex V

The curve of the intersection is a pair of straight lines passing through the vertex, and the lines are

Case II

If the plane does not pass through the vertex V

The curve of intersection is called

Hyperbola

xii.    Hyperbola

xiii.    We may obtain the section as a pair of straight lines, a circle, a parabola, an ellipse or a hyperbola, depending on the different positions of the plane cutting the cone. These curves of intersection are called conic sections.