Analytical Geometry

Chapter 15

Cone

15.1  DEFINITION OF CONE

A cone is a surface generated by a straight line which passes through a fixed point and intersects a fixed curve or touches a given curve.

The fixed point is called the vertex of the cone and the fixed curve is called a guiding curve of the cone. The straight line is called a generator.

15.2  EQUATION OF A CONE WITH A GIVEN VERTEX AND A GIVEN GUIDING CURVE

Let (α, β, γ) be the given vertex and ax² + 2hxy + by² + 2gx + 2fy + c = 0, z = 0 be the guiding curve.

The equations of any line passing through the point (α, β, γ) are

When this line meets the plane at z = 0 we get,

This point lies on the given curve ...