document.write( "Question 150278: Find the equation of the circle described each.\r
\n" ); document.write( "\n" ); document.write( "a)The circle is tangent to both coordinate axes and contains the point (6,3).\r
\n" ); document.write( "\n" ); document.write( "b)The circle is circumscribed about the triangle whose vertices are (-1,-3),(-2,4),and (2,1).\r
\n" ); document.write( "\n" ); document.write( "c)The sides of a triangle are on the line 6x+7y+11=0,2x-9y+11=0,and 9x+2y-11=0.Find the equation of the cirlce inscribed in the triangle.\r
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Algebra.Com's Answer #110375 by Edwin McCravy(20065)\"\" \"About 
You can put this solution on YOUR website!
Find the equation of the circle described each.
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document.write( "First plot the point (6,3)\r\n" );
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document.write( "I can see how there could be two different solutions, by drawing in\r\n" );
document.write( "these:\r\n" );
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document.write( "The equation of a circle with center (h,k) and radius r is\r\n" );
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document.write( "\"%28x+-+h%29%5E2+%2B+%28y-k%29%5E2+=+r%5E2\"\r\n" );
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document.write( "Draw in radii to the axes:\r\n" );
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document.write( "We can see that since the circle has to be tangent to both axes, \r\n" );
document.write( "its center has to have the same x and y coordinates. and also that\r\n" );
document.write( "the radius has to be equal to h as well.  So we can see that all\r\n" );
document.write( "three values h, k, and r, must all be the same. So let them all be \r\n" );
document.write( "h, i.e., h = k = r, and we have\r\n" );
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document.write( "So since \"h=k=r\", we have\r\n" );
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document.write( "\"%28x+-+h%29%5E2+%2B+%28y-k%29%5E2+=+r%5E2\"\r\n" );
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document.write( "\"%28x+-+h%29%5E2+%2B+%28y-h%29%5E2+=+h%5E2\"\r\n" );
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document.write( "Now since it contains the point (6,3) we can substitute that in\r\n" );
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document.write( "\"%286+-+h%29%5E2+%2B+%283-h%29%5E2+=+h%5E2\"\r\n" );
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document.write( "\"%286-h%29%286-h%29+%2B+%283-h%29%283-h%29+=+h%2A2\"\r\n" );
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document.write( "\"%2836-6h-6h%2Bh%5E2%29%2B%289-3h-3h%2Bh%5E2%29+=+h%5E2\"\r\n" );
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document.write( "\"%2836-12h%2Bh%5E2%29%2B%289-6h%2Bh%5E2%29+=+h%5E2\"\r\n" );
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document.write( "\"36-12h%2Bh%5E2%2B9-6h%2Bh%5E2+=+h%5E2\"\r\n" );
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document.write( "That simplifies to\r\n" );
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document.write( "\"h%5E2-18h%2B45=0\"\r\n" );
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document.write( "Factoring:\r\n" );
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document.write( "\"%28h-3%29%28h-15%29=0\"\r\n" );
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document.write( "\"h-3=0\" or \"h=3\"\r\n" );
document.write( "\"h-15=0\" or \"h=15\"\r\n" );
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document.write( "So we two values of \"h\", so\r\n" );
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document.write( "the two circles' equations are\r\n" );
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document.write( "\"%28x+-+3%29%5E2+%2B+%28y-3%29%5E2+=+3%5E2\" and \"%28x+-+15%29%5E2+%2B+%28y-15%29%5E2+=+15%5E2\"\r\n" );
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document.write( "\"%28x+-+3%29%5E2+%2B+%28y-3%29%5E2+=+9\" and \"%28x+-+15%29%5E2+%2B+%28y-15%29%5E2+=+225\"\r\n" );
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document.write( "b)The circle is circumscribed about the triangle whose vertices are (-1,-3),(-2,4),and (2,1). \r\n" );
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document.write( "We plot the points:\r\n" );
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document.write( "We draw the triangle:\r\n" );
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document.write( "The equation of a circle with center (\"h\",\"k\") and radius \"r\" is:\r\n" );
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document.write( "\"%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2\"\r\n" );
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document.write( "We substitute the point (\"x\",\"y\") = (\"-1\",\"-3\") \r\n" );
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document.write( "\"%28-1-h%29%5E2%2B%28-3-k%29%5E2=r%5E2\"\r\n" );
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document.write( "\"%28-1-h%29%28-1-h%29%2B%28-3-k%29%28-3-k%29=r%5E2\"\r\n" );
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document.write( "\"%281%2Bh%2Bh%2Bh%5E2%29%2B%289%2B3k%2B3k%2Bk%5E2%29=r%5E2\"\r\n" );
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document.write( "\"%281%2B2h%2Bh%5E2%29%2B%289%2B6k%2Bk%5E2%29=r%5E2\"\r\n" );
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document.write( "\"1%2B2h%2Bh%5E2%2B9%2B6k%2Bk%5E2=r%5E2\"\r\n" );
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document.write( "\"10%2B2h%2Bh%5E2%2B6k%2Bk%5E2=r%5E2\"\r\n" );
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document.write( "Get all squared terms on right:\r\n" );
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document.write( "\"10%2B2h%2B6k=r%5E2-h%5E2-k%5E2\"\r\n" );
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document.write( "We substitute the point (\"x\",\"y\") = (\"-2\",\"4\") \r\n" );
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document.write( "\"%28-2-h%29%5E2%2B%284-k%29%5E2=r%5E2\"\r\n" );
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document.write( "\"%28-2-h%29%28-2-h%29%2B%284-k%29%284-k%29=r%5E2\"\r\n" );
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document.write( "\"%284%2B2h%2B2h%2Bh%5E2%29%2B%2816-4k-4k%2Bk%5E2%29=r%5E2\"\r\n" );
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document.write( "\"%284%2B4h%2Bh%5E2%29%2B%2816-8k%2Bk%5E2%29=r%5E2\"\r\n" );
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document.write( "\"4%2B4h%2Bh%5E2%2B16-8k%2Bk%5E2=r%5E2\"\r\n" );
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document.write( "\"20%2B4h%2Bh%5E2-8k%2Bk%5E2=r%5E2\"\r\n" );
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document.write( "Get all squared terms on right:\r\n" );
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document.write( "\"20%2B4h-8k=r%5E2-h%5E2-k%5E2\"\r\n" );
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document.write( "We substitute the point (\"x\",\"y\") = (\"2\",\"1\") \r\n" );
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document.write( "\"%282-h%29%5E2%2B%281-k%29%5E2=r%5E2\"\r\n" );
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document.write( "\"%282-h%29%282-h%29%2B%281-k%29%281-k%29=r%5E2\"\r\n" );
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document.write( "\"%284-2h-2h%2Bh%5E2%29%2B%281-k-k%2Bk%5E2%29=r%5E2\"\r\n" );
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document.write( "\"%284-4h%2Bh%5E2%29%2B%281-2k%2Bk%5E2%29=r%5E2\"\r\n" );
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document.write( "\"4-4h%2Bh%5E2%2B1-2k%2Bk%5E2=r%5E2\"\r\n" );
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document.write( "\"5-4h%2Bh%5E2-2k%2Bk%5E2=r%5E2\"\r\n" );
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document.write( "Get all squared terms on right:\r\n" );
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document.write( "\"5-4h-2k=r%5E2-h%5E2-k%5E2\"\r\n" );
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document.write( "So we have the three equations:\r\n" );
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document.write( "\"10%2B2h%2B6k=r%5E2-h%5E2-k%5E2\"\r\n" );
document.write( "\"20%2B4h-8k=r%5E2-h%5E2-k%5E2\"\r\n" );
document.write( " \"5-4h-2k=r%5E2-h%5E2-k%5E2\"\r\n" );
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document.write( "Since the right sides of all three equations are equal,\r\n" );
document.write( "then so are the left sides:\r\n" );
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document.write( "\"10%2B2h%2B6k=20%2B4h-8k=5-4h-2k\"\r\n" );
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document.write( "Using the first two\r\n" );
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document.write( "\"10%2B2h%2B6k=20%2B4h-8k\"\r\n" );
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document.write( "\"-2h%2B14k=10\"\r\n" );
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document.write( "Using the first and third:\r\n" );
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document.write( "\"10%2B2h%2B6k=5-4h-2k\"\r\n" );
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document.write( "\"6h%2B8k=-5\"\r\n" );
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document.write( "So we solve these two equations by \r\n" );
document.write( "substitution of elimination:\r\n" );
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document.write( "\"-2h%2B14k=10\"\r\n" );
document.write( "\"6h%2B8k=-5\"\r\n" );
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document.write( "and get (h,k) = (\"-3%2F2\",\"1%2F2\")\r\n" );
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document.write( "To find r we go back to\r\n" );
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document.write( "\"10%2B2h%2B6k=r%5E2-h%5E2-k%5E2\"\r\n" );
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document.write( "\"10%2B2%28-3%2F2%29%2B6%281%2F2%29=r%5E2-%28-3%2F2%29%5E2-%281%2F2%29%5E2\"\r\n" );
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document.write( "\"10-3%2B3=r%5E2-%289%2F4%29-%281%2F4%29\"\r\n" );
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document.write( "\"10=r%5E2-10%2F4\"\r\n" );
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document.write( "\"10=r%5E2-5%2F2\"\r\n" );
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document.write( "\"20=2r%5E2-5\"\r\n" );
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document.write( "\"25=2r%5E2\"\r\n" );
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document.write( "\"25%2F2=r%5E2\"\r\n" );
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document.write( "\"sqrt%2825%2F2%29=+r\"\r\n" );
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document.write( "Therefore the equation \r\n" );
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document.write( "\"%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2\"\r\n" );
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document.write( "becomes\r\n" );
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document.write( "\"%28x-%28-1%2F2%29%29%5E2%2B%28y-3%2F2%29%5E2=%28sqrt%2825%2F2%29%29%5E2\"\r\n" );
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document.write( "\"%28x%2B1%2F2%29%5E2%2B%28y-3%2F2%29%5E2=25%2F2\"\r\n" );
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document.write( "So we plot the center (\"-3%2F2\",\"1%2F2\")\r\n" );
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document.write( "Now put the point of the compass on the center \r\n" );
document.write( "and draw the circle:\r\n" );
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document.write( "-----------------------\r\n" );
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document.write( "Maybe I'll do the last one tomorrow.  I'm getting sleepy. Check back to\r\n" );
document.write( "see if I've done it.\r\n" );
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document.write( "Edwin
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\n" ); document.write( "\n" ); document.write( "c)The sides of a triangle are on the line 6x+7y+11=0,2x-9y+11=0,and 9x+2y-11=0.Find the equation of the cirlce inscribed in the triangle.
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