document.write( "Question 81934: Identify the conic (circle, ellipse, or hyperbola) and state the center\r
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Algebra.Com's Answer #60090 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"2+%2A+x+%5E+2+-+y+%5E+2+%2B+2+%2A+x+%2B+1+0+%2A+y+-+4+1++++=0\"$ Start with the given conic\r
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\n" ); document.write( "\n" ); document.write( " $\"2+%2A+x+%5E+2+-+y+%5E+2+%2B+2+%2A+x+%2B+1+0+%2A+y+=+4+1+++\"$ Add 41 to both sides\r
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\n" ); document.write( "\n" ); document.write( " $\"%28+2+%2A+x+%5E+2+%2B+2+%2A+x+%29+%2B+%28+-+y+%5E+2+%2B+1+0+%2A+y+%29+=+4+1++\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2+%28+x+%5E+2+%2B+x+%29+-+1+%28+y+%5E+2+-+1+0+%2A+y+%29+=+4+1++++\"$ Factor a 2 out of the first parenthesis
\n" ); document.write( "and factor a -1 out of the second parenthesis.\r
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\n" ); document.write( "\n" ); document.write( "Now we must complete the individual squares inside the parenthesis:\r
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\n" ); document.write( "\n" ); document.write( " For instance to complete $\"2+%28+x+%5E+2+%2B+x+%29++\"$ , take half of 1 and square it (ie $\"%281%2F2%29%5E2=%280.5%29%5E2=0.25\"$ to get 0.25. Now add 0.25 inside the parenthesis like this: \r
\n" ); document.write( "\n" ); document.write( " $\"2+%28+x+%5E+2+%2B+x+%2B+0+.+2+5+%29++\"$ \r
\n" ); document.write( "\n" ); document.write( "Since you really added $\"2%2A0.25\"$ to the entire left side, we must add $\"2%2A0.25\"$ to the right side also. Now lets complete $\"-+1+%28+y+%5E+2+-+1+0+%2A+y+%29+\"$ \r
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\n" ); document.write( "\n" ); document.write( "In order to complete $\"-+1+%28+y+%5E+2+-+1+0+%2A+y+%29+\"$ , take half of -10 and square it (ie $\"%28-10%2F2%29%5E2=%28-5%29%5E2=25\"$ to get 25. Now add 25 inside the parenthesis like this: \r
\n" ); document.write( "\n" ); document.write( " $\"-+1+%28+y+%5E+2+-+1+0+%2A+y+%2B+2+5+%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "Since you really added $\"-1%2A25\"$ to the entire left side, we must add $\"-1%2A25\"$ to the right side also.\r
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Complete the individual squares by taking half of the 2nd coefficient and squaring it (remember to add to both sides).\r
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\n" ); document.write( "\n" ); document.write( " $\"2+%28+x+%5E+2+%2B+x+%2B+0+.+2+5+%29+-+1+%28+y+%5E+2+-+1+0+%2A+y+%2B+2+5+%29+=+1+6+.+5++++\"$ Combine like terms on the right side\r
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\n" ); document.write( "\n" ); document.write( " $\"2+%28+x+%2B+0+.+5+%29+%5E+2+-+1+%28+y+-+5+%29+%5E+2+=+1+6+.+5++\"$ Factor the individual groups on the left side\r
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Divide both sides by 16.5\r
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\n" ); document.write( "\n" ); document.write( " $\"1+%28+x+%2B+0+.+5+%29+%5E+2+%2F+8+.+2+5+%2B+1+%28+y+-+5+%29+%5E+2+%2F+-+1+6+.+5+=+1+\"$ Break up the fraction into standard form $\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"$ \r
\n" ); document.write( "\n" ); document.write( "So the equation is a hyperbola where the center is (-0.5,5) \r
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\n" ); document.write( "\n" ); document.write( "From the graph, you can see the center (blue dot) \r
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\n" ); document.write( "\n" ); document.write( "Here's the graph of $\"1+%28+x+%2B+0+.+5+%29+%5E+2+%2F+8+.+2+5+%2B+1+%28+y+-+5+%29+%5E+2+%2F+-+1+6+.+5+=+1+\"$ \r
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note: the straight lines are not part of the graph. They are the asymptotes with slopes of ą $\"b%2Fa\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Here is a check/verification graph of $\"2+%2A+x+%5E+2+-+y+%5E+2+%2B+2+%2A+x+%2B+1+0+%2A+y+-+4+1++++=0\"$ \r
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