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Recognising the equation of a circle.

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Which of the following are equations of a circle?

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$\\simplify[all]{(x-{ccx})^2+(y-{ccy})^2={rs[0]}}$

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$\\simplify[all]{x^2+(y-{ccy})^2={rs[1]}}$

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$\\simplify[all]{(x-{ccx})^2={rs[2]}}$

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$\\simplify[all]{{xmult}(x-{ccx})^2+{ymult}(y-{ccy})^2={rs[3]}}$

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$\\simplify[all]{{ax}x^2+{bx}x+{ax}y^2+{by}y={rs[4]}}$

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$\\simplify[all]{{ax}x^2+{bx}x+{ay}y^2+{by}y={rs[5]}}$

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$\\simplify[all]{x^2+{bx}x-y^2+{by}y={rs[5]}}$

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The standard form equation of a circle is $(x-a)^2+(y-b)^2=r^2$. It represents a circle of radius $r$ with centre $(a,b)$.

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If an equation is written in the form $cx^2+dx+ey^2+fy+g=0$, where $c,d,e,f,g$ are numbers, it could be a circle, but it might be something else (for example, an ellipse, a hyperbola, or even an empty graph). If the coefficients of $x^2$ and $y^2$ are the same then the equation is of a circle. These values being the same allow us to divide by that coefficient and start completing the square to get the equation in to the standard form of a circle. 

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