<p>Expanding \(\dfrac{(x-a)^2}{p^2}+\dfrac{(y-b)^2}{q^2}=1\) gives&nbsp;\(dx^2+ey^2+fx+gy+h=0\). Completing the square reverses the conversion.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="280" data-height="220" style="width:280px;height:220px;max-width:100%;display:block;margin:0 auto;"></div><div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--panel" data-content-box-preset="panel" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F3F6FA);color:var(--content-box-foreground, #111827);padding:10px 11px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-top-width:2px"><p style="text-align: center">\(q&gt;p\)</p></div></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="280" data-height="220" style="width:280px;height:220px;max-width:100%;display:block;margin:0 auto;"></div><div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--panel" data-content-box-preset="panel" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F3F6FA);color:var(--content-box-foreground, #111827);padding:10px 11px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-top-width:2px"><p style="text-align: center">\(q&lt;p\)</p></div></div></div></div></div>
<p>Both forms describe an ellipse with centre \((a,b)\).</p>

Expanding \(\dfrac{(x-a)^2}{p^2}+\dfrac{(y-b)^2}{q^2}=1\) gives \(dx^2+ey^2+fx+gy+h=0\). Completing the square reverses the conversion.

Both forms describe an ellipse with centre \((a,b)\).

Graphing Techniques

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Ellipses

Ellipses

Video Solution:

Solution:

Solution locked