Chapter 6: Analytic Geometry
6.1 Circles and Parabolas
6.2 Ellipses and Hyperbolas
6.3 Summary of the Conic Sections
6.4 Parametric Equations
6.1 Circles and Parabolas
• Conic Sections
– Parabolas, circles, ellipses, hyperbolas
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6.1 Circles
A circle is a set of points in a plane that are equidistant
from a fixed point. The distance is called the radius of
the circle, and the fixed point is called the center.
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A circle with center (h, k) and radius r has
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2
length r = ( x − h) + ( y − k ) to some point
(x, y) on the circle.
Squaring both sides yields the centerradius form of the equation of a circle.
r 2 = ( x − h) 2 + ( y − k ) 2
6.1 Finding the Equation of a Circle
Example
Find the center-radius form of the equation
of a circle with radius 6 and center (–3, 4). Graph the
circle and give the domain and range of the relation.
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Solution
Substitute h = –3, k = 4, and r = 6 into the
equation of a circle.
62 = ( x − ( −3))2 + ( y − 4) 2
36 = ( x + 3) 2 + ( y − 4) 2
6.1 Graphing Circles with the Graphing
Calculator
Example Use the graphing calculator to graph
the circle in a square viewing window.
x2 + y 2 = 9
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Solution
x2 + y 2 = 9
y 2 = 9 − x2
y = ± 9 − x2
Let y1 = 9 − x 2 and y2 = − 9 − x 2 .
6.1
Equations and Graphs of Parabolas
A parabola is a set of points in a plane equidistant
from a fixed point and a fixed line. The fixed point
is called the focus, and the fixed line, the directrix.
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6.1 Parabola with a Vertical Axis
The parabola with focus (0, c) and directrix y = –c has
equation x2 = 4cy. The parabola has vertical axis x = 0,
opens upward if c > 0, and opens downward if c < 0.
6.1 Parabola with a Horizontal Axis
The parabola with focus (c, 0) and directrix x = –c
has equation y2 = 4cx. The parabola has horizontal
axis y = 0, opens to the right if c > 0, and to the left
if c < 0.
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6.1 Determining Information about
Parabolas from Equations
Example Find the focus, directrix, vertex, and
axis
of each parabola.
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(b) y 2 = −28 x
(a) x = 8 y
Solution
(a) 4c = 8
c=2
Since the x-term is squared, the
parabola is vertical, with focus
at (0, c) = (0, 2) and directrix
y = –2. The vertex is (0, 0), and
the axis is the y-axis.
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6.1 Determining Information about
Parabolas from Equations
(b) 4c = −28
c = −7
The parabola is horizontal,
with focus (–7, 0), directrix
x = 7, vertex (0, 0), and
x-axis as axis of the parabola.
Since c is negative, the graph
opens to the left.
6.1 Writing Equations of Parabolas
Example
Write an equation for the parabola with
vertex (1, 3) and focus (–1, 3).
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Solution
Focus lies left of the vertex implies the
parabola has
- a horizontal axis, and
- opens to the left.
Distance between vertex and
focus is 1–(–1) = 2, so c = –2.
( y − 3) 2 = 4(−2)( x − 1)
( y − 3) 2 = −8( x − 1)
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