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      <text>6.1.18 (When do we have a circle?)</text>
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      <text><![CDATA[<h1>When is a circle not a circle?</h1>

<p>We have previously introduced the centre-radius form and the expanded form for the equation of a circle. Both of these forms are ways to represent a circle in the Cartesian plane. However, in some cases even when we have an equation in expanded form it may not actually represent a circle. In other words there are some restrictions on the expanded form in order for it to represent a circle.</p>

<p>Consider an equation in expanded form: \(x^2 + y^2 + 2gx + 2fy + c = 0\). We have seen that in this case the centre is given by \((-g,-f)\) and the radius is \(\sqrt{g^2 + f^2 -c}\). However we have previously seen (in week 1) that we can not always take the square root of a number. We require the number under the square root to be positive in order to obtain a real root. This means that in order for us to know that the expanded form actually represents a circle we require that \(g^2 + f^2 - c > 0\). Note that:</p>

<ol>
<li>if \(g^2 + f^2 - c < 0\) then we cannot obtain a real value for the radius, since we would have to square root a negative.</li>
<li>if \(g^2 + f^2 - c = 0 \) then the radius is zero, in other words the equation represents a point rather than a circle.</li>
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