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325 lines
16 KiB
HTML
325 lines
16 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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<title>Circle Equations</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Circle Equations</h1>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../geometry/images/circle.svg" alt="circle" height="225" width="212"></p>
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<p>A <a href="../geometry/circle.html">circle</a> is easy to make:</p>
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<p class="center larger"><i>Draw a curve that is "radius" away<br>
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from a central point.</i></p>
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<p>And so:</p>
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<p class="center large">All points are the same distance<br>
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from the center.</p>
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<div style="clear:both"></div>
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<p> </p>
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<p>In fact <b>the definition</b> of a circle is</p>
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<div class="def">
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<p><b>Circle:</b> The <a href="../sets/set-of-points.html">set of all points</a> on a plane that are a fixed distance from a center.</p>
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</div>
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<h2>Circle on a Graph</h2>
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<p>Let us put a circle of radius 5 on a graph:</p>
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<p class="center"><img src="images/graph-circle-5.svg" alt="graph circle" height="224" width="259"></p>
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<p>Now let's work out <b>exactly</b> where all the points are.</p>
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<p>We make a right-angled triangle:</p>
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<p class="center"><img src="images/graph-circle-5a.svg" alt="graph circle" height="224" width="259"></p>
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<p>And then use <a href="../pythagoras.html">Pythagoras</a>:</p>
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<p class="center large" style="display: block;">x<sup>2</sup> + y<sup>2</sup> = 5<sup>2</sup></p>
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<p class="larger">There are an <a href="../numbers/infinity.html">infinite</a> number of those points, here are some examples:</p>
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<p class="center"><img src="images/graph-circle-5b.svg" alt="graph circle" height="224" width="259"></p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th align="center">x</th>
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<th align="center">y</th>
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<th align="right">x<sup>2</sup> + y<sup>2</sup></th>
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</tr>
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<tr>
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<td style="text-align:center; width:30px;">5</td>
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<td style="text-align:center; width:30px;">0</td>
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<td style="text-align:right;">5<sup>2</sup> + 0<sup>2</sup> = 25 + 0 = 25</td>
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</tr>
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<tr>
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<td style="text-align:center;">3</td>
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<td style="text-align:center;">4</td>
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<td style="text-align:right;">3<sup>2</sup> + 4<sup>2</sup> = 9 + 16 = 25</td>
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</tr>
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<tr>
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<td style="text-align:center;">0</td>
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<td style="text-align:center;">5</td>
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<td style="text-align:right;">0<sup>2</sup> + 5<sup>2</sup> = 0 + 25 = 25</td>
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</tr>
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<tr>
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<td style="text-align:center;">−4</td>
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<td style="text-align:center;">−3</td>
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<td style="text-align:right;">(−4)<sup>2</sup> + (−3)<sup>2</sup> = 16 + 9 = 25</td>
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</tr>
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<tr>
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<td style="text-align:center;">0</td>
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<td style="text-align:center;">−5</td>
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<td style="text-align:right;">0<sup>2</sup> + (−5)<sup>2</sup> = 0 + 25 = 25</td>
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</tr>
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</tbody></table>
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</div>
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<p class="larger">In all cases a point on the circle follows the rule x<sup>2</sup> + y<sup>2</sup> = radius<sup>2</sup></p>
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<p>We can use that idea to find a missing value</p>
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<div class="example">
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<h3>Example: <b>x</b> value of 2, and a <b>radius</b> of 5</h3>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">x<sup>2</sup> + y<sup>2</sup> = r<sup>2</sup></span></div>
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<div class="row"><span class="left">Values we know:</span><span class="right">2<sup>2</sup> + y<sup>2</sup> = 5<sup>2</sup></span></div>
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<div class="row"><span class="left">Rearrange:</span><span class="right"> y<sup>2</sup> = 5<sup>2</sup> − 2<sup>2</sup></span></div>
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<div class="row"><span class="left">Square root both sides:</span><span class="right"> y = ±√(5<sup>2</sup> − 2<sup>2</sup>)</span></div>
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<div class="row"><span class="left">Solve:</span><span class="right">y = ±√21</span></div>
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<div class="row"><span class="left"> </span><span class="right">y ≈ <b>±4.58...</b></span></div>
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</div>
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<p><i>(The <b>±</b> means there are two possible values: one with <b>+</b> the other with <b>−</b>)</i></p>
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<p>And here are the two points:</p>
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<p class="center"><img src="images/graph-circle-5c.svg" alt="graph circle" height="224" width="259"></p>
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</div>
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<h2>More General Case</h2>
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<p>Now let us put the center at <b>(a,b)</b></p>
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<p class="center"><img src="images/graph-circle-a.svg" alt="graph circle" height="224" width="259"></p>
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<p>So the circle is <b>all the points (x,y)</b> that are <b>"r"</b> away from the center <b>(a,b)</b>.</p>
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<div style="clear:both"></div>
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<p>Now lets work out where the points are (using a right-angled triangle and <a href="../pythagoras.html">Pythagoras</a>):</p>
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<p class="center"><img src="images/graph-circle-b.svg" alt="graph circle" height="224" width="259"></p>
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<p>It is the same idea as before, but we need to subtract <b>a</b> and <b>b</b>:</p>
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<div class="def">
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<p class="center large" style="display: block;">(x−a)<sup>2</sup> + (y−b)<sup>2</sup> = r<sup>2</sup></p>
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</div>
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<p class="center larger">And that is the <b>"Standard Form"</b> for the equation of a circle!</p>
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<p> </p>
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<p>It shows all the important information at a glance: the center <b>(a,b)</b> and the radius <b>r</b>.</p>
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<div class="example">
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<h3>Example: A circle with center at (3,4) and a radius of 6:</h3>
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<p>Start with:</p>
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<p><span class="center large" style="display: block;">(x−a)<sup>2</sup> + (y−b)<sup>2</sup> = r<sup>2</sup></span></p>
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<p>Put in (a,b) and r:</p>
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<p><span class="center large" style="display: block;">(x−3)<sup>2</sup> + (y−4)<sup>2</sup> = 6<sup>2</sup></span></p>
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<p>We can then use our algebra skills to simplify and rearrange that equation, depending on what we need it for.</p>
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</div>
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<h2>Try it Yourself</h2>
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<div class="script" style="height: 360px;">
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images/circle-equn.js
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</div>
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<h2>"General Form"</h2>
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<p>But you may see a circle equation and <b>not know it</b>!</p>
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<p class="center larger">Because it may not be in the neat "Standard Form" above.</p>
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<p>As an example, let us put some values to a, b and r and then expand it</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">(x−a)<sup>2</sup> + (y−b)<sup>2</sup> = r<sup>2</sup></span></div>
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<div class="row"><span class="left">Example: a=1, b=2, r=3:</span><span class="right">(x−1)<sup>2</sup> + (y−2)<sup>2</sup> = 3<sup>2</sup></span></div>
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<div class="row"><span class="left">Expand: </span><span class="right">x<sup>2</sup> − 2x + 1 + y<sup>2</sup> − 4y + 4 = 9</span></div>
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<div class="row"><span class="left">Gather <a href="like-terms.html">like terms</a>:</span><span class="right">x<sup>2</sup> + y<sup>2</sup> − 2x − 4y + 1 + 4 − 9 = 0</span></div>
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</div>
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<p>And we end up with this:</p>
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<p class="center"><span class="large">x<sup>2</sup> + y<sup>2</sup> − 2x − 4y − 4 = 0</span></p>
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<p class="center">It is a circle equation, but "in disguise"!</p>
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<p>So when you see something like that think <i>"hmm ... that <b>might</b> be a circle!"</i></p>
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<p>In fact we can write it in <b>"General Form"</b> by putting constants instead of the numbers:</p>
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<div class="def">
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<p class="center"><span class="large">x<sup>2</sup> + y<sup>2</sup> + Ax + By + C = 0</span></p>
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</div>
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<p><i>Note: General Form always has <span class="large">x<sup>2</sup> + y<sup>2</sup></span> for the first two terms</i>.</p>
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<h2>Going From General Form to Standard Form</h2>
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<p>Now imagine we have an equation in <b>General Form</b>:</p>
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<p class="center large">x<sup>2</sup> + y<sup>2</sup> + Ax + By + C = 0</p>
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<p>How can we get it into <b>Standard Form</b> like this?</p>
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<p class="center large">(x−a)<sup>2</sup> + (y−b)<sup>2</sup> = r<sup>2</sup></p>
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<p>The answer is to <a href="completing-square.html">Complete the Square</a> (read about that) twice ... once for <b>x</b> and once for <b>y</b>:</p>
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<div class="example">
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<h3>Example: x<sup>2</sup> + y<sup>2</sup> − 2x − 4y − 4 = 0</h3>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">x<sup>2</sup> + y<sup>2</sup> − 2x − 4y − 4 = 0</span></div>
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<div class="row"><span class="left">Put <b>x</b>s and <b>y</b>s together:</span><span class="right"><span class="larger">(x<sup>2</sup> − 2x) + (y<sup>2</sup> − 4y) − 4 = 0</span></span></div>
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<div class="row"><span class="left">Constant on right:</span><span class="right"><span class="larger">(x<sup>2</sup> − 2x) + (y<sup>2</sup> − 4y) = 4</span></span></div>
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</div>
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<p>Now complete the square for <b>x</b> (take half of the −2, square it, and add to both sides):</p>
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<p class="larger center">(x<sup>2</sup> − 2x + <span class="hilite">(−1)<sup>2</sup></span>) + (y<sup>2</sup> − 4y) = 4 + <span class="hilite">(−1)<sup>2</sup></span></p>
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<p>And complete the square for <b>y</b> (take half of the −4, square it, and add to both sides):</p>
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<p class="larger center">(x<sup>2</sup> − 2x + (−1)<sup>2</sup>) + (y<sup>2</sup> − 4y + <span class="hilite">(−2)<sup>2</sup></span>) = 4 + (−1)<sup>2</sup> + <span class="hilite">(−2)<sup>2</sup></span></p>
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<p>Tidy up:</p>
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<div class="tbl">
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<div class="row"><span class="left">Simplify:</span><span class="right">(x<sup>2</sup> − 2x + 1) + (y<sup>2</sup> − 4y + 4) = 9</span></div>
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<div class="row"><span class="left">Finally:</span><span class="right">(x − 1)<sup>2</sup> + (y − 2)<sup>2</sup> = 3<sup>2</sup></span></div>
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</div>
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<p>And we have it in Standard Form!</p>
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<p>(Note: this used the <span class="left"> a=1, b=2, r=3 </span>example from before, so we got it right!)</p>
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</div>
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<h2>Unit Circle</h2>
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<p>If we place the circle center at (0,0) and set the radius to 1 we get:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="../geometry/images/circle-unit-pythagoras.gif" alt="Unit Circle" height="219" width="221"></td>
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<td>
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<p class="center larger">(x−a)<sup>2</sup> + (y−b)<sup>2</sup> = r<sup>2</sup></p>
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<p class="center larger">(x−0)<sup>2</sup> + (y−0)<sup>2</sup> = 1<sup>2</sup></p>
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<p class="center"><span class="large">x<sup>2</sup> + y<sup>2</sup> = 1</span></p>
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Which is the equation of the <a href="../geometry/unit-circle.html">Unit Circle</a></td>
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</tr>
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</tbody></table>
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<h2>How to Plot a Circle by Hand</h2>
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<p>1. Plot the center <b>(a,b)</b></p>
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<p>2. Plot 4 points "radius" away from the center in the up, down, left and right direction</p>
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<p>3. Sketch it in!</p>
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<div class="example">
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<h3>Example: Plot (x−4)<sup>2</sup> + (y−2)<sup>2</sup> = 25</h3>
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<p>The formula for a circle is <span class="large">(x−a)<sup>2</sup> + (y−b)<sup>2</sup> = r<sup>2</sup></span></p>
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<p>So the center is at <span class="large">(4,2)</span></p>
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<p>And <b>r<sup>2</sup></b> is <b>25</b>, so the radius is <span class="large">√25 = 5</span></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/graph-circle-c.gif" alt="graph circle" height="231" width="260"></p>
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<p>So we can plot:</p>
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<ul>
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<li>The Center: (4,2)</li>
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<li>Up: (4,2+5) = (4,7)</li>
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<li>Down: (4,2−5) = (4,−3)</li>
|
||
<li>Left: (4−5,2) = (−1,2)</li>
|
||
<li>Right: (4+5,2) = (9,2)</li>
|
||
</ul><div style="clear:both"></div>
|
||
<p>Now, just sketch in the circle the best we can!</p>
|
||
</div>
|
||
|
||
|
||
<h2>How to Plot a Circle on the Computer</h2>
|
||
|
||
<p>We need to rearrange the formula so we get "y=".</p>
|
||
<p>We should end up with two equations (top and bottom of circle) that can then be plotted.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Plot (x−4)<sup>2</sup> + (y−2)<sup>2</sup> = 25</h3>
|
||
<p>So the center is at <span class="large">(4,2)</span>, and the radius is <span class="large">√25 = 5</span></p>
|
||
<p>Rearrange to get "y=":</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"> (x−4)<sup>2</sup> + (y−2)<sup>2</sup> = 25</span></div>
|
||
<div class="row"><span class="left">Move (x−4)<sup>2</sup> to the right:</span><span class="right"> (y−2)<sup>2</sup> = 25 − (x−4)<sup>2</sup></span></div>
|
||
<div class="row"><span class="left">Take the square root: </span><span class="right">(y−2) = ± √[25 − (x−4)<sup>2</sup>]</span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><i>(notice the ± "plus/minus" ...<br>
|
||
there can be two square roots!)</i></span></div>
|
||
<div class="row"><span class="left">Move the "−2" to the right:</span><span class="right">y = 2 ± √[25 − (x−4)<sup>2</sup>]</span></div>
|
||
</div>
|
||
<p> </p>
|
||
<p>So when we plot these two equations we should have a circle:</p>
|
||
<ul>
|
||
<li><span class="larger">y = 2 + √[25 − (x−4)<sup>2</sup>]</span></li>
|
||
<li><span class="larger">y = 2 − √[25 − (x−4)<sup>2</sup>]</span></li>
|
||
</ul>
|
||
<p>Try plotting those functions on the <a href="../data/function-grapher.html">Function Grapher</a>.</p>
|
||
<p>It is also possible to use the <a href="../data/grapher-equation.html">Equation Grapher</a> to do it all in one go.</p>
|
||
</div>
|
||
|
||
<p> </p>
|
||
<div class="questions">8526, 8527, 8539, 8540, 8515, 8516, 569, 8544, 8559, 8560, 570, 1209</div>
|
||
|
||
<div class="related">
|
||
<a href="../geometry/circle.html">Circle</a>
|
||
<a href="../geometry/unit-circle.html">Unit Circle</a>
|
||
<a href="../geometry/index.html">Geometry Index</a>
|
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<a href="index.html">Algebra Index</a>
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