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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