The document explores different types of conic sections, including ellipses, hyperbolas, circles, and parabolas. It shows how to use completing the square to rewrite general form conic equations into standard form equations for each type. The values of coefficients A, B, C, D, E determine whether the conic is an ellipse, hyperbola, circle, or may degenerate into a line or point. When B is not equal to 0, polar coordinates can be used to graph the conic section.

1. Exploring Conic Sections
Any conic section from ellipse and circle to parabola and hyperbola
may be described by the implicit relation:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
(1) Consider the case when (A ≠ C) are positive coefficients and B=0.
(a) Use the technique of completing the square to rewrite the general
form conic:
4x2 + 0xy + 9y2 + 8x - 18y - 23 = 0
as a standard form ellipse:
(x-h)2 (y-k)2
+ = 1.
a2 b2
(b) Solve this conic for y as an explicit function in x, y = f(x) and graph in
function mode.
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2. Exploring Conic Sections
(2) Consider the case when (A ≠ C) are coefficients with opposite signs
and B=0.
(a) Use the technique of completing the square to rewrite the general
form conic:
2x2 + 0xy - 8y2 - 8x + 16y - 16 = 0
as a standard form hyperbola:
(x-h)2 (y-k)2
- = 1.
a2 b2
(b) Solve this conic for y as an explicit function in x, y = f(x) and graph in
function mode.
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3. Exploring Conic Sections
(3) Consider the case when (A = C) and B=0.
(a) Use the technique of completing the square to rewrite the general
form conic:
2x2 + 0xy + 2y2 + 8x + 4y + 2 = 0
as a standard form circle:
(x - h)2 + (y - k)2 = r2
(b) Solve this conic for y as an explicit function in x, y = f(x) and graph in
function mode.
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4. Exploring Conic Sections
(4) Summarize what you discovered about conic sections so far.
(a) What is the significance of h and k?
(b) What do the vales of D and E affect?
(c) What do the values of a, b and r represent?
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5. Exploring Conic Sections
(5) Consider the general form conic with B≠0:
x2 - xy + y2 + 0x + 0y - 12 = 0
(a) Substitute x=rcos(θ), y=rsin(θ) and solve algebraically for r as an
explicit function of θ, r = f(θ).
(b) Graph your f(θ) using polar mode.
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6. Exploring Conic Sections
(6) Consider the general form conic with B≠0:
x2 + 2xy + y2 + 0x + 0y - 12 = 0
(a) Substitute x=rcos(θ), y=rsin(θ) and solve algebraically for r as an
explicit function of θ, r = f(θ).
(b) Graph your f(θ) using polar mode.
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7. Exploring Conic Sections
(7) Consider the general form conic with B≠0:
x2 + 4xy + y2 + 0x + 0y - 12 = 0
(a) Substitute x=rcos(θ), y=rsin(θ) and solve algebraically for r as an
explicit function of θ, r = f(θ).
(b) Graph your f(θ) using polar mode.
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8. Exploring Conic Sections
(8) Extend what you have learned about conic sections!
(a) Calculate B2-4AC for each conic graphed above. What do you
think the value of B2-4AC determines?
(b) What effect did B≠0 have?
(c) What about the black sheep of the family? Sometimes conic
sections degenerate to lines. When does this occur?
(d) Change one coefficient to make
2x2 + 2y2 + 8x + 4y = -2
degenerate to a point!
(e) Given that the standard form parabola is written:
(x - k)2 = 4p(y - h) vert symmetry, vertex (h, k), focus (h, k + p),
(y - k)2 = 4p(x - h) hort symmetry, vertex (h, k), focus (h + p, k),
how can you make a general form conic represent a parabola?
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