document.write( "Question 1077688: An ellipse and a hyperbola have the same foci, A and B, and intersect at four points. The ellipse has major axis 50, and minor axis 40. The hyperbola has conjugate axis of length 20. Let $P$ be a point on both the hyperbola and ellipse. What is PA*PB? \n" ); document.write( "

Algebra.Com's Answer #692209 by KMST(5328) $\"\"$ $\"About$

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The ellipse has
\n" ); document.write( " $\"a=50%2F2=25\"$ (the semi-major axis), and
\n" ); document.write( " $\"b=40%2F2=20\"$ (the semi-minor axis).
\n" ); document.write( "The focal distance, $\"c\"$ , can be found from
\n" ); document.write( " $\"b%5E2%2Bc%5E2=a%5E2\"$ .
\n" ); document.write( "Substituting known values, $\"20%5E2%2Bc%5E2=25%5E2\"$ --> $\"c%5E2=625-400=225\"$ --> $\"c=15\"$ .
\n" ); document.write( "So, it looks like this

(maybe shifted and/or rotated).
\n" ); document.write( "According to the definition of ellipse,
\n" ); document.write( "if P is a point of an ellipse with foci A and B,
\n" ); document.write( " $\"PA%2BPB=2a=50\"$ .
\n" ); document.write( "
\n" ); document.write( "The hyperbola has
\n" ); document.write( " $\"c=15\"$ , just like the ellipse,
\n" ); document.write( "and has $\"b=20%2F2=10\"$ .
\n" ); document.write( "Knowing that in a hyperbola
\n" ); document.write( " $\"a%5E2%2Bb%5E2=c%5E2\"$ , we can find $\"a\"$ .
\n" ); document.write( " $\"a%5E2%2B15%5E2=10%5E2\"$
\n" ); document.write( " $\"a%5E2%2B225=100\"$
\n" ); document.write( " $\"a%5E2=225-100=125\"$
\n" ); document.write( " $\"a=sqrt%28125%29\"$
\n" ); document.write( "In a hyperbola with foci A and B, for any point P, by definition
\n" ); document.write( " $\"abs%28PA-PB%29=2a=2sqrt%28125%29\"$ .
\n" ); document.write( "
\n" ); document.write( "Squaring both sides in the equations found involving the distances PA and PB,
\n" ); document.write( " $\"%28PA%2BPB%29%5E2=2500\"$ --> $\"PA%5E2%2BPB%5E2%2B2%2APA%2APB=2500\"$ and
\n" ); document.write( " $\"%28PA-PB%29%5E2=500\"$ ---> $\"PA%5E2%2BPB%5E2-2%2APA%2APB=500\"$ .
\n" ); document.write( "Subtracting $\"PA%5E2%2BPB%5E2-2%2APA%2APB=500\"$ from $\"PA%5E2%2BPB%5E2%2B2%2APA%2APB=2500\"$ , we get
\n" ); document.write( " $\"PA%5E2%2BPB%5E2%2B2%2APA%2APB-PA%5E2-PB%5E2%2B2%2APA%2APB=2500-500\"$ --> $\"4%2APA%2APB=2000\"$ --> $\"PA%2APB=2000%2F4=highlight%28500%29\"$ \n" ); document.write( "

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