Circle equation vs. semicircle equation

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In summary, for the equation of a circle with a radius of 1, the equation x^2 + y^2 = 1 represents a full circle, while the rearranged equation y= √(1 - x^2) only represents a semicircle. This is because the two equations are not equivalent and the first one has more solutions than the second one. The positive square root represents the upper half circle, while the negative square root represents the lower half circle.
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Lo.Lee.Ta.
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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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  • #2
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!
 

Related to Circle equation vs. semicircle equation

What is the difference between a circle equation and a semicircle equation?

A circle equation represents the set of all points that are equidistant from a fixed point, known as the center. A semicircle equation represents half of a circle, with the center being the endpoint of the diameter.

What is the general form of a circle equation?

The general form of a circle equation is (x - h)2 + (y - k)2 = r2, where (h, k) represents the coordinates of the center and r represents the radius.

How do you graph a circle or semicircle using an equation?

To graph a circle or semicircle using an equation, first identify the center and radius. Then, plot the center point on a coordinate plane and use the radius to mark points around the center, creating a circular or semicircular shape.

What are the important properties of a circle and semicircle?

The important properties of a circle and semicircle include the radius, diameter, circumference, and area. The radius is the distance from the center to any point on the circle. The diameter is the distance across the circle, passing through the center. The circumference is the distance around the circle, and the area is the space inside the circle.

How are circle and semicircle equations used in real life?

Circle and semicircle equations have various real-life applications, such as in architecture for designing circular structures, in engineering for creating circular or semicircular components, and in physics for calculating the motion of objects in circular paths. They are also used in fields like optics, astronomy, and geography to analyze and measure circular phenomena.

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