§ 1

In a Cartesian rectangular coordinate system the points $A(1,0)$ and $B(4,0)$ are given. Write the equation of the set of points that are twice as far from $B$ as from $A$.

Each of the two equations of the system $(x-2)^{2}+y^{2}=r^{2}$, $(x+2)^{2}+y^{2}=r^{2}$ defines a circle in a Cartesian rectangular coordinate system. Subtracting one equation from the other, we obtain a consequence of this system: $x=0$. How is this result to be interpreted geometrically? Consider the cases $r=3$ and $r=1$.

Form the equation of the cylinder whose directrix is given by the system of equations

and whose generators are parallel to the vector $\mathbf{e}_{3}$.

Write the equation of the cone with vertex at the origin and with directrix given by the system of equations $x^{2}+y^{2}=4, z=1$.

What is the distinguishing feature of the equation of a cone in a spherical coordinate system whose center is at the vertex of the cone?