4.4 Conics • Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas. • Write the equation and find the focus of a parabola. • Write the equation of a ellipse and find the foci, vertices, the length of the major and minor
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4.4 Conics Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas. Write the equation and find the focus.
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4.4 Conics
• Recognize the equations and graph the four basic conics: parabolas, circles, ellipse, and hyperbolas.
• Write the equation and find the focus of a
parabola.
• Write the equation of a ellipse and find the
foci, vertices, the length of the major and minor
axis.
Parabolas
Definition: A parabola is the set of all points (x, y) in a plane that are equidistant from a fixed line, called the directrix, and a fixed point, the focus, not on the line.
Directrix
Parabolas
x2 = 4py p 0Vertex (0, 0)Directrix y = -pFocus (0, p)Line of sym x = 0
y2 = 4px p 0Vertex (0,0)Directrix x = -pFocus (p, 0)Line of sym y = 0
Standard equation of the Parabola
Parabola ExamplesGiven 22xy Find the focus. Since the squared variable is x, the
parabola is oriented in the y directions. The leading coefficient is negative, so, the parabola opens down.
-pp
Focus (0, p)
•
Solve for x2
pyyx
yx
42
12
1
2
2
Solve for p.
p
p
8
1
42
1
Focus )8
1,0(
Parabola ExamplesWrite the standard form of the equation of the parabola with the vertex at the origin and the focus (2, 0).
Note that the focus is along the x axis, so the parabola is oriented in the x axis direction, y2 = 4px.