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2x^2 + 2y^2 – 8x + 10y + 2 =0 PLEASE help me with this problem. Rewrite in standard form. Find the center and radius of the circle. Sh
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2x^2 + 2y^2 – 8x + 10y + 2 =0
PLEASE help me with this problem. Rewrite in standard form. Find the center and radius of the circle. Show all of your work.
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Mathematics
5 months
2021-08-10T20:35:33+00:00
2021-08-10T20:35:33+00:00 1 Answers
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Answers ( )
A circle of radius r centered at (a, b) has equation
(x – a)² + (y – b)² = r ²
To get the given equation in this form, complete the square with respect to both x and y.
2x² – 8x = 2 (x² – 4x)
2x² – 8x = 2 (x² – 4x + 4 – 4)
2x² – 8x = 2 ((x – 2)² – 4)
2x² – 8x = 2 (x – 2)² – 8
2y² + 10y = 2 (y² + 5y)
2y² + 10y = 2 (y² + 5y + 25/4 – 25/4)
2y² + 10y = 2 ((y + 5/2)² – 25/4)
2y² + 10y = 2 (y + 5/2)² – 25/2
So we have
2x² + 2y² – 8x + 10y + 2 = 0
2 (x – 2)² – 8 + 2 (y + 5/2)² – 25/2 + 2 = 0
2 (x – 2)² + 2 (y + 5/2)² = 37/2
(x – 2)² + (y + 5/2)² = 37/4
(x – 2)² + (y + 5/2)² = ((√37)/2)²
which means this circle has radius (√37)/2 and center (2, -5/2).