362CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES
Theorem 9.1.2 Suppose that v(h, k) is the vertex and the line y = k − p is the directrix of a parabola. Then the focus is F (h, k + p) and the axis is the vertical line with equation x = h. The equation of the parabola is
(x − h)2 = 4p(y − k).
9.2Ellipse
Definition 9.2.1 An ellipse is the locus of all points, the sum of whose distances from two fixed points, called foci, is a fixed positive constant that is greater than the distance between the foci. The midpoint of the line segment joining the two foci is called the center. The line segment through the foci and with end points on the ellipse is called the major axis. The line segment, through the center, that has end points on the ellipse and is perpendicular to the major axis is called the minor axis. The intersections of the major and minor axes with the ellipse are called the vertices.
Theorem 9.2.1 Let an ellipse have center at (h, k), foci at (h ± c, k), ends
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a, k |
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and ends of the minor axis at (h, k |
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b), where |
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of the major axis at ( |
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a > 0, b > 0, c > 0 and a |
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(x − h)2 |
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(y − k)2 |
= 1. |
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The length of the major axis is 2a and the length of the minor axis is 2b.
Theorem 9.2.2 Let an ellipse have center at (h, k), foci at (h, k ± c), ends
h, k |
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a and the ends of the minor axis at (h |
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b, k), |
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of the major axis at ( |
± ), 2 |
= b |
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where a > 0, b > 0, c > 0 and a |
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+ c |
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is |
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(y − k)2 |
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(x − h)2 |
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= 1. |
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a2 |
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The length of the major axis is 2a and the length of the minor axis is 2b.
Remark 24 If c = 0, then a = b, foci coincide with the center and the ellipse reduces to a circle.
9.3. HYPERBOLA |
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9.3Hyperbola
Definition 9.3.1 A hyperbola is the locus of all points, the di erence of whose distances from two fixed points, called foci, is a fixed positive constant that is less than the distance between the foci. The mid point of the line segment joining the two foci is called the center. The line segment, through the foci, and with end points on the hyperbola is called the major axis. The end points of the major axis are called the vertices.
Theorem 9.3.1 Let a hyperbola |
have |
center at |
(h, k), |
foci at (h ± c, k), |
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vertices at (h ± a, k), where 0 < a < c, |
b = |
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c2 − a2 |
, then the equation of |
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the hyperbola is |
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(x − h)2 |
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(y − k)2 |
= 1. |
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Theorem 9.3.2 Let a hyperbola |
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center at |
(h, k), |
foci at (h, k ± c), |
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vertices at (h, k ± a), where 0 < a < c, |
b = |
√ |
c2 − a2 |
, then the equation of |
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the hyperbola is |
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(y − k)2 |
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(x − h)2 |
= 1. |
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9.4Second-Degree Equations
Definition 9.4.1 The transformations
x = x0 cos θ − y0 sin θ y = x0 sin θ + y0 cos θ
and
x0 = x cos θ + y sin θ y0 = −x sin θ + y cos θ
are called rotations. The point P (x, y) has coordinates (x0, y0) in an x0y0- coordinate system obtained by rotating the xy-coordinate system by an angle
θ.
Theorem 9.4.1 Consider the equation ax2 + bxy + cy2 + dx + ey + f = 0, b 6= 0. Let cot 2θ = (a − c)/b and x0y0-coordinate system be obtained