Find the Properties (x^2)/2+((y-8)^2)/64=1

Math
x22+(y-8)264=1
Simplify each term in the equation in order to set the right side equal to 1. The standard form of an ellipse or hyperbola requires the right side of the equation be 1.
x22+(y-8)264=1
This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.
(x-h)2b2+(y-k)2a2=1
Match the values in this ellipse to those of the standard form. The variable a represents the radius of the major axis of the ellipse, b represents the radius of the minor axis of the ellipse, h represents the x-offset from the origin, and k represents the y-offset from the origin.
a=8
b=2
k=8
h=0
The center of an ellipse follows the form of (h,k). Substitute in the values of h and k.
(0,8)
Find c, the distance from the center to a focus.
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Find the distance from the center to a focus of the ellipse by using the following formula.
a2-b2
Substitute the values of a and b in the formula.
(8)2-(2)2
Simplify.
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Raise 8 to the power of 2.
64-(2)2
Rewrite 22 as 2.
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Use axn=axn to rewrite 2 as 212.
64-(212)2
Apply the power rule and multiply exponents, (am)n=amn.
64-212⋅2
Combine 12 and 2.
64-222
Cancel the common factor of 2.
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Cancel the common factor.
64-222
Divide 1 by 1.
64-21
64-21
Evaluate the exponent.
64-1⋅2
64-1⋅2
Simplify the expression.
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Multiply -1 by 2.
64-2
Subtract 2 from 64.
62
62
62
62
Find the vertices.
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The first vertex of an ellipse can be found by adding a to k.
(h,k+a)
Substitute the known values of h, a, and k into the formula.
(0,8+8)
Simplify.
(0,16)
The second vertex of an ellipse can be found by subtracting a from k.
(h,k-a)
Substitute the known values of h, a, and k into the formula.
(0,8-(8))
Simplify.
(0,0)
Ellipses have two vertices.
Vertex1: (0,16)
Vertex2: (0,0)
Vertex1: (0,16)
Vertex2: (0,0)
Find the foci.
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The first focus of an ellipse can be found by adding c to k.
(h,k+c)
Substitute the known values of h, c, and k into the formula.
(0,8+62)
The first focus of an ellipse can be found by subtracting c from k.
(h,k-c)
Substitute the known values of h, c, and k into the formula.
(0,8-(62))
Simplify.
(0,8-62)
Ellipses have two foci.
Focus1: (0,8+62)
Focus2: (0,8-62)
Focus1: (0,8+62)
Focus2: (0,8-62)
Find the eccentricity.
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Find the eccentricity by using the following formula.
a2-b2a
Substitute the values of a and b into the formula.
(8)2-(2)28
Simplify the numerator.
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Raise 8 to the power of 2.
64-(2)28
Rewrite 22 as 2.
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Use axn=axn to rewrite 2 as 212.
64-(212)28
Apply the power rule and multiply exponents, (am)n=amn.
64-212⋅28
Combine 12 and 2.
64-2228
Cancel the common factor of 2.
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Cancel the common factor.
64-2228
Divide 1 by 1.
64-218
64-218
Evaluate the exponent.
64-1⋅28
64-1⋅28
Multiply -1 by 2.
64-28
Subtract 2 from 64.
628
628
628
These values represent the important values for graphing and analyzing an ellipse.
Center: (0,8)
Vertex1: (0,16)
Vertex2: (0,0)
Focus1: (0,8+62)
Focus2: (0,8-62)
Eccentricity: 628
Find the Properties (x^2)/2+((y-8)^2)/64=1

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