document.write( "Question 390113: If a circle intersects the hyperbola \"y+=+1%2Fx\" at four distinct points (x_i, y_i), i = 1,2,3,4, then prove that x_1*x_2 = y_3*y_4. \n" ); document.write( "
Algebra.Com's Answer #276617 by robertb(5830)\"\" \"About 
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We have to solve the system
\n" ); document.write( "\"%28x-h%29%5E2+%2B+%28y-+k%29%5E2+=+r%5E2\",
\n" ); document.write( "\"y+=+1%2Fx\".
\n" ); document.write( "In the equation of the circle above, h, k, and r are all constants. The problem also assures us that the circle and the hyperbola will intersect in 4 distinct points.
\n" ); document.write( "Substitute the bottom equation into the top equation: \"%28x-h%29%5E2+%2B+%281%2Fx-+k%29%5E2+=+r%5E2\".
\n" ); document.write( "Expand the resulting equation:
\n" ); document.write( "\"x%5E2+-+2hx+%2B+h%5E2+%2B+1%2Fx%5E2+-+%282k%29%2Fx+%2B+k%5E2+=+r%5E2\".
\n" ); document.write( "Clear fractions:
\n" ); document.write( "\"x%5E4+-+2hx%5E3+%2B+h%5E2x%5E2+%2B+1+-+2kx+%2B+k%5E2x%5E2+-+r%5E2x%5E2+=+0\".
\n" ); document.write( "Combine like terms:
\n" ); document.write( "\"x%5E4+-+2hx%5E3+%2B+%28h%5E2+%2B+k%5E2+-+r%5E2%29x%5E2+-+2kx+%2B+1+=+0\" <----(A)\r
\n" ); document.write( "\n" ); document.write( "From algebra, we know that for the polynomial
\n" ); document.write( " \"a%5B0%5Dx%5En+%2B+a%5B1%5Dx%5E%28n-1%29\" + ...+ \"a%5Bk%5D%2Ax%5E%28n-k%29\" +...+ \"a%5Bn-2%5Dx%5E2+%2B+a%5Bn-1%5Dx+%2B+a%5Bn%5D+=+0\", \r
\n" ); document.write( "\n" ); document.write( "\"%28-1%29%5Ek%2Aa%5Bk%5D+\" \r
\n" ); document.write( "\n" ); document.write( "is equal to the summation of product of the roots of the polynomial taken k at a time. We need this result only for the constant term of (A). We get \"%28-1%29%5E4x%5B1%5D%2Ax%5B2%5D%2Ax%5B3%5D%2Ax%5B4%5D+=+1\".
\n" ); document.write( "This is the same as \"x%5B1%5D%2Ax%5B2%5D%2A%281%2Fy%5B3%5D%29%2A%281%2Fy%5B4%5D%29+=+1\", or finally,
\n" ); document.write( "\"x%5B1%5D%2Ax%5B2%5D+=+y%5B3%5D%2Ay%5B4%5D\". \n" ); document.write( "
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