Chapter 9 Study Guide:
1. Know the Distance Formula and the Midpoint formula
Sample Problems:
(a) Find the distance and the midpoint between points (-3, 1) and (-7, 5).
(b) Find the distance and the midpoint between (1/2, 0) and (3, -6).
2. Know the formula for a circle and how to apply it.
Sample Problems:
(a) Find an equation of the circle with center (3, -4) and radius 6.
(b) Find the center and radius of the circle x^2 + y^2 – 6x + 2y + 6 = 0. Then graph.
(c) A circle with center at the origin is translated four units to the left and 5 units up. What is
the new center of the circle?
Remember that some circles have no solution. (Look back at your notes to remember a case like this).
3. Know the formulas for a parabola (both orientations) and how to apply them.
Sample Problems:
(a) A parabola has focus (1, 0) and directrix y= 3. Find its vertex and state the equation.
(b) Find the axis, vertex, focus, and directrix of the parabola x^2 – 2x – 2y = 0. Sketch the
parabola.
(c) Find an equation of the parabola that has vertex (3, 0) and focus (3, -3).
4. Know the formulas for an ellipse (both orientations) and how to apply them.
Sample Problems:
(a) Give the x- and y-intercepts and find the foci of the ellipse 9x^2 + 4y^2 = 144. State the
major axis.
(b) Graph the ellipse x^2 + 4y^2 – 16 = 0 and find the foci.
5. Know the formulas for a hyperbola (both orientations) and how to apply them.
Sample Problems:
(a) For the hyperbola 16x^2 = 9y^2 + 144, sketch the graph, showing asymptotes as dashed
lines. Also find the foci.
(b) Graph the parabola y^2 = 9x^2 + 9. Show asymptotes and find the foci.
6. Know how to identify a conic based on a general equation and know how to find the equation for
and graph a conic section whose center is (h, k) and not (0,0).
Sample Problems:
(a) Identify the conic. Then, find its center and foci, and graph. 4x^2 + 9y^2 – 16x + 18y – 11 = 0
(b) Identify the conic, find its center and foci, then graph: 16x^2 – 9y^2 + 64x + 18y + 199 = 0.
7. Know how to solve a system of three equations with three variables.
Sample Problems:
(a) x + y -3z = 10
y + z = 12
z = -2
(b) 2x + y + 3z = 10
x - 2y + z = 10
-4x + 3y + 2z = 5