document.write( "Question 1183327: A circle is tangent to the line 3x-4y -4 = 0 at the point (-4,-4) and the center is on the line
\n" ); document.write( "x+y+7 = 0. Find the equation of the circle
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Algebra.Com's Answer #813570 by robertb(5830) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let $\"%28x-x%5B0%5D%29%5E2%2B%28y-y%5B0%5D%29%5E2+=+d%5E2\"$ be the equation of the circle. \r
\n" ); document.write( "\n" ); document.write( "Note that the point ( $\"x%5B0%5D\"$ , $\"y%5B0%5D\"$ ) is on the line x + y + 7 = 0, hence $\"x%5B0%5D+%2B+y%5B0%5D+%2B+7+=+0\"$ , or $\"+y%5B0%5D+=+-x%5B0%5D+-+7\"$ .\r
\n" ); document.write( "\n" ); document.write( "Since the point (-4,-4) is on the circle, then the the equation \r
\n" ); document.write( "\n" ); document.write( " $\"%28-4-x%5B0%5D%29%5E2%2B%28-4-y%5B0%5D%29%5E2+=+d%5E2\"$ is true. This becomes $\"%28x%5B0%5D%2B4%29%5E2%2B%28y%5B0%5D%2B4%29%5E2+=+d%5E2\"$ after simplifying.\r
\n" ); document.write( "\n" ); document.write( "Also, $\"%28x%5B0%5D%2B4%29%5E2%2B%28-x%5B0%5D+-+7+%2B4%29%5E2+=+d%5E2\"$ , or $\"%28x%5B0%5D%2B4%29%5E2%2B%28x%5B0%5D+%2B3%29%5E2+=+d%5E2\"$ after the fact $\"+y%5B0%5D+=+-x%5B0%5D+-+7\"$ .\r
\n" ); document.write( "\n" ); document.write( " $\"%28x%5B0%5D%2B4%29%5E2%2B%28x%5B0%5D+%2B3%29%5E2+=+d%5E2\"$ <==> $\"2x%5B0%5D%5E2%2B14x%5B0%5D%2B25+=+d%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Since the point (-4,-4) is on the line 3x - 4y - 4 = 0, and the radial line is perpendicular to this line,
\n" ); document.write( "we can find the distance between 3x - 4y - 4 = 0 and the center ( $\"x%5B0%5D\"$ , $\"y%5B0%5D\"$ ).
\n" ); document.write( "The distance is given by

.\r
\n" ); document.write( "\n" ); document.write( "==> $\"d%5E2+=+%287x%5B0%5D%2B24%29%5E2%2F25+=+%2849x%5B0%5D%5E2+%2B+336x%5B0%5D+%2B+576%29%2F25\"$ \r
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\n" ); document.write( "\n" ); document.write( "==> $\"2x%5B0%5D%5E2%2B14x%5B0%5D%2B25+=+%2849x%5B0%5D%5E2+%2B+336x%5B0%5D+%2B+576%29%2F25\"$ , or\r
\n" ); document.write( "\n" ); document.write( " $\"50x%5B0%5D%5E2+%2B350x%5B0%5D%2B+625+=+49x%5B0%5D%5E2+%2B+336x%5B0%5D+%2B+576\"$ , or $\"x%5B0%5D%5E2+%2B+14x%5B0%5D+%2B49+=+0\"$ , or $\"%28x%5B0%5D+%2B+7%29%5E2+=+0\"$ ==> $\"x%5B0%5D+=+-7\"$ .\r
\n" ); document.write( "\n" ); document.write( "Then $\"+y%5B0%5D+=+-x%5B0%5D+-+7\"$ ==> $\"y%5B0%5D+=+0\"$ . Therefore the center is (-7,0).\r
\n" ); document.write( "\n" ); document.write( "Also, $\"d%5E2+=+%287x%5B0%5D%2B24%29%5E2%2F25+=+%287%2A-7%2B24%29%5E2%2F25+=+25\"$ .\r
\n" ); document.write( "\n" ); document.write( "Therefore the equation of the circle is $\"highlight%28%28x%2B7%29%5E2%2By%5E2+=+25%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "This is practically an engineering problem, and some other tutor here might have a much shorter solution.
\n" ); document.write( "(Admittedly I'm working from memory on this one.)
\n" ); document.write( "From what I remember there is an even shorter formula for a problem like this, but I cannot retrieve that formula.\r
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