Section 10.6: Translating
Conic Sections
What You’ll Learn: to translate conic
sections and write and identify the equation
of translated conic section
Why: to use location navigation systems
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Writing Equations of Translated
Conic Sections
Conic
Section
Standard Form of Equation
Parabola
Vertex (0,0)
y = ax2
x = ay2
Vertex (h,k)
y-k=a(x-h)2 or y=a(x-h)2+k
x-k=a(y-h)2 or x=a(y-h)2+k
Circle
Center (0,0)
x2 + y 2 = r 2
Center (h,k)
(x-h)2 + (y-k)2 = r2
Ellipse
Center (0,0)
x2/a2 + y2/b2 = 1
x2/b2 + y2/a2 = 1
Center (h,k)
(x-h)2/a2 + (y-k)2/b2 = 1
(x-h)2/b2 + (y-k)2/a2 = 1
Hyperbola
Center (0,0)
x2/a2 - y2/b2 = 1
y2/b2 - x2/a2 = 1
Center (h,k)
(x-h)2/a2 - (y-k)2/b2 = 1
(y-k)2/b2 + (x-h)2/a2 = 1
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Example 1
 Write the equation of each conic section:
 Ellipse with center (-3,-2); vertical major axis of
length 8; minor axis of length 6.
 Hyperbola with vertices (0,1) and (6,1) and foci
(-1,1) and (7,1)
 Translate each of the following situations 3
units up and 2 units left.
 x2/4 – y2/16 = 1
 x2 + y2 = 9
 x2/4 + y2/16 = 1
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Example 2
 Identify the conic section with equation
4x2 + y2 – 24x + 6y + 9 = 0
 Identify the conic section represented by
each equation:
 x2 + 14x – 4y + 29 = 0
 x2 + y2 – 12x + 4y = 8
 Describe the translation that would
produce the equation
x2 – 2y2 + 6x – 7 = 0
Homework
Section 10-6 HW pages 495-496: 1-21 (no
graphs)
Any questions that involve writing or
explaining should be done in complete
sentences and show critical thinking skills.
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