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Unformatted text preview: r = 1−eed θ) and r = 1+eed θ) , respectively. The key thing to
remember is that in any of these cases, the directrix is always perpendicular to the major axis of
an ellipse and it is always perpendicular to the transverse axis of the hyperbola. For parabolas,
knowing the focus is (0, 0) and the directrix also tells us which way the parabola opens. We have
established the following theorem.
6 Turn r = e(d + r cos(θ)) into r = e(d + x) and square both sides to get r2 = e2 (d + x)2 . Replace r2 with x2 + y 2 ,
expand (d + x)2 , combine like terms, complete the square on x and clean things up.
Since e > 1 in this case, 1 − e2 < 0. Hence, we rewrite 1 − e2 = e2 − 1 to help simplify things later on. 836 Applications of Trigonometry Theorem 11.12. Suppose e and d are positive numbers. Then
• the graph of r = ed
1−e cos(θ) is the graph of a conic section with directrix x = −d. • the graph of r = ed
1+e cos(θ) is the graph of a conic section with directrix x = d. • the graph of r = ed
1−e sin(θ) is the graph of a conic section with directrix...
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