Equation of circle in quarter/half of a circle

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SUMMARY

The equation of a quarter circle, represented as y = ±√(r² - x²), is fundamentally the same as that of a half circle, but with specific restrictions on the x-values. For the upper right quarter circle, the equation is y = +√(r² - x²) with the restriction 0 ≤ x ≤ r. Conversely, the upper left quarter circle has the restriction -r ≤ x ≤ 0. The half circle encompasses both positive and negative values of y, with the restriction -r ≤ x ≤ r, while the lower half and quarter circles utilize the negative square root.

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Homework Statement



I am curious why is the equation of a quarter of a circle (y = \pm \sqrt{r^{2}-x^2}) the same as half a circle? Shouldnt they be different?

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The Attempt at a Solution



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If you want quarter circle you have to use either + (for positive quarter) and - (for negative quarter) in the equation you mentioned.
For getting half circle both + and - ve values of equation are considered.
 
TsAmE said:

Homework Statement



I am curious why is the equation of a quarter of a circle (y = \pm \sqrt{r^{2}-x^2}) the same as half a circle? Shouldnt they be different?
Actually, they are different if you include restrictions on x. For example, the equation for the upper right quarter circle is
y = +\sqrt{r^{2}-x^2}, 0 \le x \le r

The equation for the upper left quarter circle has a different restriction on x; namely
-r \le x \le 0

For the upper half of the circle, you have -r \le x \le r

For the lower half circle and quarter circles, the only difference is that the negative square root is used.
 

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