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Unformatted text preview: y = −d. • the graph of r = ed
1+e sin(θ) is the graph of a conic section with directrix y = d. In each case above, (0, 0) is a focus of the conic and the number e is the eccentricity of the conic.
• If 0 < e < 1, the graph is an ellipse whose major axis has length
has length √2ede2
1−e2 and whose minor axis • If e = 1, the graph is a parabola whose focal diameter is 2d.
• If e > 1, the graph is a hyperbola whose transverse axis has length
axis has length √2ed 1 .
e2 − 2ed
e2 −1 and whose conjugate We test out Theorem 11.12 in the next example.
Example 11.6.4. Sketch the graphs of the following equations.
1. r = 4
1 − sin(θ) 2. r = 12
3 − cos(θ) 3. r = 6
1 + 2 sin(θ) Solution.
1. From r = 1−sin(θ) , we ﬁrst note e = 1 which means we have a parabola on our hands. Since
ed = 4, we have d = 4 and considering the form of the equation, this puts the directrix
at y = −4. Since the focus is at (0, 0), we know that the vertex is located at the point
(in rectangular coor...
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