Geometry: An Interactive Journey to Mastery

Lesson 29: Folding and Conics

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and no sides congruent?

2. $FLUFXODUF\OLQGHULVVOLFHGDWDQDQJOHͼ௘6HHFigure 29.13௘ͽ,VWKHRYDOVKDSHG
   curve produced an ellipse?


3. In Lesson 20, we saw that the equation of a circle can be written as x^2 + y^2 = r^2.
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   This equation can be rewritten as follows.
   xrr 222  y^2 1.
   We also hinted at the end of that lesson that the graph of a more general
   equation of the form abx^222  y^21 is an oval shape that looks like an ellipse.
   Prove that this is indeed the case. Do this by showing that if an ellipse is centered about the origin and has
   foci on the xD[LVDWͼ௘íc௘ͽDQGͼ௘c௘ͽIRUH[DPSOHWKHQDQ\SRLQWͼ௘x, y௘ͽRQWKHHOOLSVHVDWLV¿HVDQHTXDWLRQ
   of the form
   xab^222  y^2 1.
   Warning: Answering this question requires hefty algebraic work!

Figure 29.10

Figure 29.11

Figure 29.12

Figure 29.13