Chapter II

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?

Find the parametric equations of the line given by the equations

Find the parametric equations of the plane $x-2 y+3 z=1$.

Find the coordinates of the point of intersection of the lines with the equations: $x=1-t, y=1+t, z=1-t$ and $x=3 t-1, y=2 t-2$, $z=1+t$. What value of the parameter corresponds to this point on each of the lines? How can one establish that the lines meet without finding the point of intersection?

Write the equation of the plane in which the lines of exercise 3 lie.

Write the parametric equations of the lines given by the vector equations

In a Cartesian rectangular coordinate system the coordinates of the vertices of a triangle are given: $A(20,-15), B(-16,0)$ and $C(-8,6)$. Find the coordinates of the center and the radius of the circle inscribed in the triangle.

The origin lies in one of the angles formed by the lines with the equations $A_{1} x+B_{1} y+C_{1}=0$ and $A_{2} x+B_{2} y+C_{2}=0$ in a Cartesian rectangular coordinate system. Under what necessary and sufficient condition on the coefficients of the equations is this angle acute?

Form the equation of the line passing through the origin and meeting the lines with the equations $x=1+2 t$, $y=2+3 t, z=-t$ and $x=4 t, y=5-5 t, z=3+2 t$.

Find the radius and the coordinates of the center of the sphere passing through the point $A(0,1,0)$ and tangent to the planes with the equations

The coordinate system is Cartesian rectangular.

In a Cartesian rectangular coordinate system the coordinates of the vertices of a triangle $A(1,2,3), B(1,5,-1)$ and $C(5,3,-5)$ are given. Find the coordinates of the center of the circle circumscribed about the triangle.

Write the equations of the line which is parallel to the line $\mathbf{r}=\mathbf{r}_{0}+\mathbf{a}_{0} t$ and meets the lines $\mathbf{r}=\mathbf{r}_{1}+\mathbf{a}_{1} t$ and $\mathbf{r}=\mathbf{r}_{2}+\mathbf{a}_{2} t$. It is given that the vectors $\mathbf{a}_{0}, \mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are not coplanar.

Write the parametric equations of the line which is the intersection of the planes with the equations $\mathbf{r}=\mathbf{r}_{1}+u \mathbf{a}+v \mathbf{b}$ and $\left(\mathbf{r}-\mathbf{r}_{2}, \mathbf{n}\right)=0$.