Find the center and radius of the circle.
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Dr.Peterson
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Presumably you mean 2x^2 + 2y^2 − x + y = 1, that is, [MATH]2x^2 + 2y^2 − x + y = 1[/MATH].
The standard way to answer the question is to complete the square on each variable; but there may be other ways. In order to help you most effectively, we'd like you to show what you have tried (by any method), so we can see what you have been taught, and where you are getting stuck.
Can you do that? This is not the easiest problem of this sort; you may find it best (again, depending on what you have been taught) to divide the equation by 2 before doing anything else.
The standard way to answer the question is to complete the square on each variable; but there may be other ways. In order to help you most effectively, we'd like you to show what you have tried (by any method), so we can see what you have been taught, and where you are getting stuck.
Can you do that? This is not the easiest problem of this sort; you may find it best (again, depending on what you have been taught) to divide the equation by 2 before doing anything else.
i) divide both sides by 2
ii) complete the square in both x and y on the left hand side
iii) transfer any constants left over from (ii) over to the right hand side
iv) read off the center as [MATH](x_c,y_c)[/MATH] by seeing [MATH](x-x_c)^2 + (y-y_c)^2[/MATH] on the left hand side
v) read off the radius as [MATH]r=\sqrt{\text{right hand side}}[/MATH]
vi) enjoy the beverage of your choice to celebrate a job well done!
ii) complete the square in both x and y on the left hand side
iii) transfer any constants left over from (ii) over to the right hand side
iv) read off the center as [MATH](x_c,y_c)[/MATH] by seeing [MATH](x-x_c)^2 + (y-y_c)^2[/MATH] on the left hand side
v) read off the radius as [MATH]r=\sqrt{\text{right hand side}}[/MATH]
vi) enjoy the beverage of your choice to celebrate a job well done!
D
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Please share your work/thoughts about this assignment.
Start with:
x2 + y2 − (1/2)x + (1/2)y = 1/2
x2 − 2*(1/4)x + (1/4)2 + y2 + 2*(1/4)y + (1/4)2 = 1/2 + (1/4)2 + (1/4)2
continue.....
If you are still not sure how to proceed here is an example:
Example:
Find the radius and centre of the circle:
4x² + 4y² - 8x + (40/3)y = 164/9
36x² + 36y² - 72x + 120y = 164
36[x² + y² - 2x + (10/3)y] = 164
18[x² + y² - 2x + (10/3)y] = 82
18[{(x² - 2x + 1) - 1}+ {(y² + (10/3)y + 25/9) - 25/9}] = 82
18[{(x - 1)² - 1} + {(y + 5/3)² - 25/9}] = 82
9[{(x - 1)² - 1} + {(y + 5/3)² - 25/9}] = 41
9[(x - 1)² + (y + 5/3)²] = 75
(x - 1)² + (y + 5/3)² = 75/9
Centre = (1,-5/3) Radius = (5√3)/3
Example:
Find the radius and centre of the circle:
4x² + 4y² - 8x + (40/3)y = 164/9
36x² + 36y² - 72x + 120y = 164
36[x² + y² - 2x + (10/3)y] = 164
18[x² + y² - 2x + (10/3)y] = 82
18[{(x² - 2x + 1) - 1}+ {(y² + (10/3)y + 25/9) - 25/9}] = 82
18[{(x - 1)² - 1} + {(y + 5/3)² - 25/9}] = 82
9[{(x - 1)² - 1} + {(y + 5/3)² - 25/9}] = 41
9[(x - 1)² + (y + 5/3)²] = 75
(x - 1)² + (y + 5/3)² = 75/9
Centre = (1,-5/3) Radius = (5√3)/3
