document.write( "Question 1019885: Obtain the equations of the circles which touch the y axis and pass through the points (2,5) and (4,3) \n" ); document.write( "
Algebra.Com's Answer #635889 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "If a circle touches the y-axis, then its radius equals\r\n" );
document.write( "the x-coordinate of the center. That is, r=h. The center (h,k)\r\n" );
document.write( "becomes (r,k).  \r\n" );
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document.write( "So the equation is of the form:\r\n" );
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document.write( "\"%28x-r%29%5E2%2B%28y-k%29%5E2=r%5E2\"\r\n" );
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document.write( "The points (2,5) and (4,3) must satisfy this equation, so we have\r\n" );
document.write( "this system of equations to solve:\r\n" );
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document.write( "\"system%28%282-r%29%5E2%2B%285-k%29%5E2=r%5E2%2C%284-r%29%5E2%2B%283-k%29%5E2=r%5E2%29\"\r\n" );
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document.write( "Working with the first equation:\r\n" );
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document.write( "\"%285-k%29%5E2=r%5E2-%282-r%29%5E2\"\r\n" );
document.write( "\"%285-k%29%5E2=r%5E2-%284-4r%2Br%5E2%29\"\r\n" );
document.write( "\"%285-k%29%5E2=r%5E2-4%2B4r-r%5E2\"\r\n" );
document.write( "\"%285-k%29%5E2=-4%2B4r\"\r\n" );
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document.write( "Working with the second equation:\r\n" );
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document.write( "\"%283-k%29%5E2=r%5E2-%284-r%29%5E2\"\r\n" );
document.write( "\"%283-k%29%5E2=r%5E2-%2816-8r%2Br%5E2%29\"\r\n" );
document.write( "\"%283-k%29%5E2=r%5E2-16%2B8r-r%5E2\"\r\n" );
document.write( "\"%283-k%29%5E2=-16%2B8r\"\r\n" );
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document.write( "We can eliminate r by multiplying the results\r\n" );
document.write( "of the first equation by -2 and adding to\r\n" );
document.write( "the results of the second equation:\r\n" );
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document.write( "\"-2%285-k%29%5E2=8-8r\"\r\n" );
document.write( "\"%283-k%29%5E2=-16%2B8r\"\r\n" );
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document.write( "\"-2%285-k%29%5E2%2B%283-k%29%5E2=-8\"\r\n" );
document.write( "\"-2%2825-10k%2Bk%5E2%29%2B%289-6k%2Bk%5E2%29=-8\"\r\n" );
document.write( "\"-50%2B20k-2k%5E2%2B9-6k%2Bk%5E2=-8\"\r\n" );
document.write( "\"-k%5E2%2B14k-33+=+0\"\r\n" );
document.write( "\"k%5E2-14k%2B33+=+0\"\r\n" );
document.write( "\"%28k-3%29%28k-11%29=0\"\r\n" );
document.write( "k=3, k=11\r\n" );
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document.write( "Substituting k=3 in\r\n" );
document.write( "\"%285-k%29%5E2=-4%2B4r\"\r\n" );
document.write( "\"%285-3%29%5E2=-4%2B4r\"\r\n" );
document.write( "\"2%5E2=-4%2B4r\"\r\n" );
document.write( "\"4=-4%2B4r\"\r\n" );
document.write( "\"8=4r\"\r\n" );
document.write( "\"2=r\"\r\n" );
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document.write( "So the small circle has \r\n" );
document.write( "center (2,3) and radius 2, and it has\r\n" );
document.write( "equation \r\n" );
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document.write( "\"%28x-2%29%5E2%2B%28y-3%29%5E2=2%5E2\"\r\n" );
document.write( "\"%28x-2%29%5E2%2B%28y-3%29%5E2=4\"\r\n" );
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document.write( "Substituting k=11 in\r\n" );
document.write( "\"%285-k%29%5E2=-4%2B4r\"\r\n" );
document.write( "\"%285-11%29%5E2=-4%2B4r\"\r\n" );
document.write( "\"%28-5%29%5E2=-4%2B4r\"\r\n" );
document.write( "\"36=-4%2B4r\"\r\n" );
document.write( "\"40=4r\"\r\n" );
document.write( "\"10=r\"\r\n" );
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document.write( "So the large circle has \r\n" );
document.write( "center (10,11) and radius 10, and it has\r\n" );
document.write( "equation \r\n" );
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document.write( "\"%28x-10%29%5E2%2B%28y-11%29%5E2=10%5E2\"\r\n" );
document.write( "\"%28x-10%29%5E2%2B%28y-11%29%5E2=100\"\r\n" );
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document.write( "Edwin
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