Circle equation vs. semicircle equation

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The equation of a circle with a radius of 1 is expressed as x² + y² = 1. When rearranged to y = √(1 - x²), it only represents the upper semicircle because it only accounts for the positive square root. The complete representation of the circle includes both the positive and negative square roots, yielding y = ±√(1 - x²), which encompasses both the upper and lower halves of the circle. Thus, the rearrangement limits the solutions to half of the original circle.

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This may sound like a dumb question, but...

For the equation of a circle with a radius of 1, the equation is:

x^2 + y^2 = 1


But if we rearrange that to be: y= √(1 - x^2),
then it's only a semicircle...

Why is that? Why is it now a semicircle just because it got rearranged?

Thanks!
 
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Lo.Lee.Ta. said:
This may sound like a dumb question, but...

For the equation of a circle with a radius of 1, the equation is:

x^2 + y^2 = 1


But if we rearrange that to be: y= √(1 - x^2),
then it's only a semicircle...
Your two equations are not equivalent, meaning that the first equation has more solutions than the second one.

If you solve for y in the first equation, you should get y = ±√(1 - x2). The pos. square root represents the upper half circle; the neg. square root represents the lower half circle.
Lo.Lee.Ta. said:
Why is that? Why is it now a semicircle just because it got rearranged?

Thanks!
 

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