a. Calculate the solution set and
b. Demonstrate that your solutions are right by drawing the graphs, preferably by computer.
$x^2+2 y^2=33$\

$$x^2+y^2+2 x=19$$

a. Complete the square (if necessary), find the center, draw the asymptotes, plot the vertices, then sketch the graph.
b. Calculate the focal radius, and plot the foci.
c. Confirm your sketch by plotting the original equation, using PLOT CONIC on the accompanying disk, or a similar program.
$x^2-y^2-14 x-8 y+37=0$

Prove that the graph of an inverse variation function is a hyperbola.

What will the graph be?\

$$x-y=25$$

Sketch the graph of:
A function with a removable discontinuity at $x=3$.

Find a polynomial equation with integer coefficients for the set of coplanar points described. tell whether or not the graph is a conic section and, if it is, tell which conic section.
Each point is three times as far from line $y=5$ as it is from line $x=1$.

Solve the system.
$7 x^2-9 y^2=31$\

$$5 x^2+9 y^2=161$$

a. Find the vertex and intercepts.
b. Sketch the graph.
c. Confirm your sketch using PLOT CONIC from the accompanying disk, or another suitable graphing program.
Find the vertex and intercepts of the following parabolas, and sketch the graph. Find decimal approximations for any radicals you encounter.\

$$x=\frac{1}{3} y^2-2 y-9$$

Solve: $\sqrt{x}+3=7$

Evaluate $\sqrt[5]{32}$

For problem:
a. Sketch the graph of the relation,
b. Calculate the focal radius and plot the two foci,
c. Confirm your sketch by plotting the original equation, using PLOT CONIC on the accompanying computer disk, or similar program.\

$$16 x^2+y^2-128 x-20 y+292=0$$

Suppose that you are aboard a spaceship. The Earth is at the origin of a Cartesian coordinate system, and the path of your spaceship is the graph of

$$9 x^2+25 y^2-72 x=81 .$$

a. Which conic section is this path, and how do you tell?
b. Sketch the graph of this path.
c. Show that the Earth is at the focus of this conic section.
d. To determine your position at a particular time, you tune in to the LORAN station and find that you are also on the graph of

$$9 x^2-15 y^2=9 .$$

Which conic section is this graph, and how do you tell?
e. By solving the system formed by the equations of the two conic sections above, find the three possible points at which your spaceship could be located.
f. Draw the graph of the conic section in part $\mathbf{d}$ on the same Cartesian coordinate system as in part b, and thus show that your answer to part e is correct.
g. To ascertain at which one of the three points you are located, you find that you are also on the graph of

$$3 x-5 y=27 .$$

At which of the points are you located? (There is a clever way to do this, as well as the long way.)

What will the graph be?\

$$x^2+5 x y-9 y^2=100$$

a. Calculate the solution set and
b. Demonstrate that your solutions are right by drawing the graphs, preferably by computer.
$9 x^2+y^2=85$\

$$2 x^2-3 y^2=6$$