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      <text>6.1.19 (Example of deciding if an equation is a circle)</text>
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      <text><![CDATA[<h3>Example</h3>

<p>Decide whether the equation \(x^2 + y^2 - 18x + 20y +364 = 0\) is the equation of a circle.</p>

<h3>Solution</h3>

<p>
We recall that the expanded form of a circle is given by \(x^2 + y^2 +2gx +2fy + c = 0\), thus comparing this with the given equation we get that 
\[
2g =-18 \implies g = -9 \qquad \mbox{and} \qquad 2f = 20 \implies f = 10 \qquad \mbox{and} \qquad c = 364.
\]
Then we calculate 
\[
g^2 + f^2 - c = (-9)^2 + 10^2 - 364 = 81 + 100 - 364 = -183 < 0,
\]
therefore we cannot obtain a real value for the radius so the given equation is not that of a circle.
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