## 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 *ax*^{2} + 2*hxy* + *by*^{2} + 2*gx* + 2*fy* + *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 ...