Name: ______________________________________________
Honors Math 2
Conic Sections Test:
Study Guide and Review Questions
Topics to know:
o Finding the distance and midpoint between two points (distance formula and midpoint formula)
o The “section of a cone” definition for each conic section
o The geometric (or distance) definition of each conic section
o Derive the algebraic standard equations from the geometric definitions for each conic section
o Graph parabolas, circles, ellipses, and hyperbolas
o Identify/graph and/or calculate all important features of each conic section:
o Parabola: vertex, focus, directrix, focal width
o Circle: center, radius, diameter, area, circumference
o Ellipse: center, vertices, co-vertices, major axis, minor axis, foci, eccentricity
o Hyperbola: center, vertices, co-vertices, transverse axis, conjugate axis, foci, equations of
asymptotes
o Solve non-linear systems using algebraic and graphical methods
o Apply your knowledge to “real-world” problems that require the use of conic sections. For example, see
review question #4
Some Review Questions:
1.
a. Describe the graph of the equation (x – 1)2 + (y + 2)2 = 4. Give enough information that someone could draw
the graph using your description.
b. Suppose a new curve was formed by taking the graph from part a and stretching it to 5 times as wide and 3
times as tall, while keeping the same center. Tell which type of curve would be obtained, and write its
equation.
c. Let F1 and F2 stand for the foci (focal points) of the curve from part b. Find the coordinates of F1 and F2.
d. On graph paper, graph the curve from part b, and draw points F1 and F2.
2. Answer the following questions about the equation –9x2 + 4y2 – 72x – 8y = 176.
a. Consider the graph of this equation. What type of curve is it?
b. Rewrite the equation in one of the standard forms for this type of curve.
c. Which of the following features does this curve have? Circle all that apply.
“center”
“one vertex”
“two vertices”
“directrix”
“two asymptotes” “one focus” “two foci”
d. Graph the curve along with all of the features you circled in part c.
“co-vertices”
3. A parabola has its vertex at V = (5, 2) and its focus at F = (1, 2).
a. Make a rough sketch of this parabola. Include the parabola, its focus, and its directrix.
Label the directrix with its equation.
b. Find an equation for this parabola, in one of the standard forms.
c. Now find the equation(s) that you could use to graph the same curve on your calculator. Show what you
would enter on the calculator. (hint: solve for y . Your calculator only accepts “y =”)
4. The planet Pluto has an elliptical orbit with the Sun at one focus. Suppose that this ellipse is placed in the coordinate
plane with the sun at (0, 0) and the other focus along the positive y-axis.
a. The distance from Pluto to the sun varies between a minimum of 2763 million miles and a maximum of 4587
million miles.
Using these facts, determine the lengths of the ellipse’s major and minor axes.
b. Write an equation for the ellipse orbit, in terms of the variables x and y. Also, identify the coordinates of the
foci.
2
2
2
5. Explain how you know that a  b  c for an ellipse. Justify your reasoning, and show a picture to support your
answer.
6. For each conic section given below, change it to standard form, identify what type of conic section it is, and then graph
each!
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a.
x 2  2x  y  3  0
b.
9x 2  16y 2  54 x  32y  47  0
c.
25y 2  9x 2  50y  54 x  281  0
d.
9x 2  4 y 2  36x  8y  4
e.
4 x 2  y 2  124  32x 16y
f.
9x 2 18x  54 y  9y 2  86
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7. Find an equation of a hyperbola that is the graph of all points P with PF1  PF2  2 2 with foci F1
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

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2 , 2 and
F2  2 , 2 . (Note that this problem can not be done using the standard equation that we derived in class because
the foci do no lie on either a horizontal or vertical axis. You must use the geometric definition to derive the equation.)