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TsAmE
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Homework Statement
I am curious why is the equation of a quarter of a circle ([tex]y = \pm \sqrt{r^{2}-x^2}[/tex]) the same as half a circle? Shouldnt they be different?
Homework Equations
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The Attempt at a Solution
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Actually, they are different if you include restrictions on x. For example, the equation for the upper right quarter circle isTsAmE said:Homework Statement
I am curious why is the equation of a quarter of a circle ([tex]y = \pm \sqrt{r^{2}-x^2}[/tex]) the same as half a circle? Shouldnt they be different?
The general equation of a circle in a quarter or half of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.
To find the center and radius, you can use the coordinates of three points on the circle. The center will be the midpoint of the line connecting any two points, and the radius will be half of the distance between the center and any of the three points.
No, the radius of a circle cannot be negative as it represents the distance from the center to any point on the circle. If the radius is negative, it would mean that the circle has imaginary points, which is not possible.
The general equation of a circle (x - h)^2 + (y - k)^2 = r^2 applies to circles in any position, while the equation of a circle in a quarter or half of a circle only applies to circles in specific positions. Additionally, the radius may be different in the two equations depending on the position of the circle.
Yes, the equation of a circle in a quarter or half of a circle can also be written as x^2 + y^2 = r^2 or y = ±√(r^2 - x^2), depending on the orientation of the circle. These forms may be more useful in certain situations, such as when graphing the circle.