document.write( "Question 163807: can you please help me solve these questions:-
\n" ); document.write( "1. \"what is the equation of the circle who touches the negative x- and y-axes and also the line\"7x+%2B24y+%2B12+=+0\"\"
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Algebra.Com's Answer #120685 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let's graph the line:\r\n" );
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document.write( "We can draw two possible circles that touch that line and also the two\r\n" );
document.write( "axes, a great big one and a tiny one: \r\n" );
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document.write( "So we expect two solutions, both in quadrant 3.  \r\n" );
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document.write( "Let the radius of the circle be r, In either case the circle's center\r\n" );
document.write( "will then be (-r,-r). \r\n" );
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document.write( "If any circle with radius r in quadrant 3 touches both axes then \r\n" );
document.write( "the points (-r,0) and (0,-r) are the points where the\r\n" );
document.write( "circle touches the axes, and the circle has center (-r,-r), and \r\n" );
document.write( "thus the equation of the circle is\r\n" );
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document.write( "\"%28x%2Br%29%5E2+%2B+%28y%2Br%29%5E2=r%5E2\"\r\n" );
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document.write( "This can be seen by drawing in these two radii,\r\n" );
document.write( "since they are also the coordinates of the center (-r,-r)\r\n" );
document.write( "(I'll just do it with the big circle.):\r\n" );
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document.write( "Now if we draw a third radius touching the line:\r\n" );
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document.write( "We know that the distance from a point \r\n" );
document.write( "\"matrix%281%2C5%2C+%22%28%22%2C+x%5B0%5D%2C++%22%2C%22%2C+y%5B0%5D%2C++%22%29%22%29\" to a\r\n" );
document.write( "line \"Ax%2BBy%2BC=0\" is found by this equation:\r\n" );
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document.write( "\"d=%28abs%28Ax%5B0%5D%2BBy%5B0%5D%2BC%29%29%2Fsqrt%28A%5E2%2BB%5E2%29\"\r\n" );
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document.write( "So the distance from the line \"7x+%2B24y+%2B12+=+0\"\r\n" );
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document.write( "to the center (-r,-r) must also be equal to the radius r,\r\n" );
document.write( "of the circle\r\n" );
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document.write( "So substituting \r\n" );
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document.write( "\"d=%28abs%28Ax%5B0%5D%2BBy%5B0%5D%2BC%29%29%2Fsqrt%28A%5E2%2BB%5E2%29\"\r\n" );
document.write( "\"r=%28abs%287%28-r%29%2B24%28-r%29%2B12%29%29%2Fsqrt%287%5E2%2B24%5E2%29\"\r\n" );
document.write( "\"r=%28abs%28-7r-24r%2B12%29%29%2Fsqrt%2849%2B576%29\"\r\n" );
document.write( "\"r=%28abs%28-31r%2B12%29%29%2Fsqrt%28625%29\"\r\n" );
document.write( "\"r=%28abs%28-31r%2B12%29%29%2F25\"\r\n" );
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document.write( "Squaring both sides\r\n" );
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document.write( "\"r%5E2=%28-31r%2B12%29%5E2%2F625\"\r\n" );
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document.write( "\"625r%5E2=%28-31r%2B12%29%5E2\"\r\n" );
document.write( "\"625r%5E2=%28-31r%2B12%29%28-31r%2B12%29\"\r\n" );
document.write( "\"625r%5E2=961r%5E2-744r%2B144\"\r\n" );
document.write( "\"0=336r%5E2-744r%2B144\"\r\n" );
document.write( "\"336r%5E2-744r%2B144=0\"\r\n" );
document.write( "Dividing through by 24\r\n" );
document.write( "\"14r%5E2-31r%2B6\"\r\n" );
document.write( "\"%28r-2%29%2814r-3%29=0\"\r\n" );
document.write( "   \r\n" );
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document.write( "So the big circle has radius 2 and its equation becomes\r\n" );
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document.write( "\"%28x-r%29%5E2%2B%28y-r%29%5E2=r%5E2\"\r\n" );
document.write( "\"%28x-2%29%5E2%2B%28y-2%29%5E2=2%5E2\"\r\n" );
document.write( "\"%28x-2%29%5E2%2B%28y-2%29%5E2=4\"\r\n" );
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document.write( "And the little circle has radius 3/14 and its equation becomes\r\n" );
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document.write( "\"%28x-r%29%5E2%2B%28y-r%29%5E2=r%5E2\"\r\n" );
document.write( "\"%28x-3%2F14%29%5E2%2B%28y-3%2F14%29%5E2=%283%2F14%29%5E2\"\r\n" );
document.write( "\"%28x-3%2F14%29%5E2%2B%28y-3%2F14%29%5E2=9%2F196\"\r\n" );
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document.write( "2.\" what is the equation of the circle who passes \r\n" );
document.write( "through the points (1,4),(7,5),(1,8)?\"\r\n" );
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document.write( "Use the general equation of a circle:\r\n" );
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document.write( "\"x%5E2%2By%5E2%2BDx%2BEy%2BF=0\"\r\n" );
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document.write( "Substitute the point (x,y) = (1,4)\r\n" );
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document.write( "\"%281%29%5E2%2B%284%29%5E2%2BD%281%29%2BE%284%29%2BF=0\"\r\n" );
document.write( "\"1%2B16%2BD%2B4E%2BF=0\"\r\n" );
document.write( "\"17%2BD%2B4E%2BF=0\"\r\n" );
document.write( "\"D%2B4E%2BF=-17\"\r\n" );
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document.write( "Substitute the point (x,y) = (7,5)\r\n" );
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document.write( "\"%287%29%5E2%2B%285%29%5E2%2BD%287%29%2BE%285%29%2BF=0\"\r\n" );
document.write( "\"49%2B25%2B7D%2B5E%2BF=0\"\r\n" );
document.write( "\"74%2B7D%2B5E%2BF=0\"\r\n" );
document.write( "\"7D%2B5E%2BF=-74\"\r\n" );
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document.write( "Substitute the point (x,y) = (1,8)\r\n" );
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document.write( "\"%281%29%5E2%2B%288%29%5E2%2BD%281%29%2BE%288%29%2BF=0\"\r\n" );
document.write( "\"1%2B64%2BD%2B8E%2BF=0\"\r\n" );
document.write( "\"65%2BD%2B8E%2BF=0\"\r\n" );
document.write( "\"D%2B8E%2BF=-65\"\r\n" );
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document.write( "Now we have a system of three equations in three\r\n" );
document.write( "unknowns:\r\n" );
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document.write( "\"system%28D%2B4E%2BF=-17%2C7D%2B5E%2BF=-74%2CD%2B8E%2BF=-65%29\"\r\n" );
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document.write( "This has solution:\r\n" );
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document.write( "\"matrix%281%2C5%2C+D=-15%2F2+%2C+%22%2C%22+%2C+E=-12%2C+%22%2C%22+%2C+F=77%2F2%29\"\r\n" );
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document.write( "So the equation of the circle:\r\n" );
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document.write( "\"x%5E2%2By%5E2%2BDx%2BEy%2BF=0\" becomes\r\n" );
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document.write( "\"x%5E2%2By%5E2-%2815%2F2%29x%2B%28-12%29y%2B%2877%2F2%29=0\"\r\n" );
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document.write( "Multiplying through by 2:\r\n" );
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document.write( "\"2x%5E2%2B2y%5E2-15x-24y%2B77=0\"\r\n" );
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document.write( "3.\"what is the equation of the circle who has its centre \r\n" );
document.write( "on the line \"x+%2B+y+=+1+\"and passes through the origin \r\n" );
document.write( "and the point (4,2)?\"\r\n" );
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document.write( "We'll draw the line and mark the point (4,2):\r\n" );
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document.write( "  \r\n" );
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document.write( "Now since the circle must go through (0,0) and (4,2),\r\n" );
document.write( "the line segment joining these two points must be a \r\n" );
document.write( "chord of the circle.  So we'll draw it.\r\n" );
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document.write( "Now the perpendicular bisector of a chord must pass through\r\n" );
document.write( "the center of the circle, so let's get the equation of the\r\n" );
document.write( "perpendicular bisector of that chord from (0,0) to (4,2).\r\n" );
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document.write( "First we need its slope.  So we find the slope of the chord,\r\n" );
document.write( "take its reciprocal with the opposite sign.\r\n" );
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document.write( "Slope of the chord:\r\n" );
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document.write( "\"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-y%5B1%5D%29\"\r\n" );
document.write( "\"m=%282-0%29%2F%284-0%29\"\r\n" );
document.write( "\"m=2%2F4\"\r\n" );
document.write( "\"m=1%2F2\"\r\n" );
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document.write( "So the slope of its perpendicular bisector is \"-2%2F1\" or \"-2\".\r\n" );
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document.write( "The perpendicular bisector must also go through the midpoint of\r\n" );
document.write( "the chord, so we use the midpoint formula:\r\n" );
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document.write( "\"M=%28matrix%281%2C3++++%2C++%280%2B4%29%2F2+%2C+%22%2C%22+%2C+%280%2B2%29%2F2+++%29%29\"\r\n" );
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document.write( "\"M=%28matrix%281%2C3++++%2C++2+%2C+%22%2C%22+%2C+1+++%29%29\"\r\n" );
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document.write( "So we find the equation of the perpendicular bisector of the\r\n" );
document.write( "chord.  It has slope = m = \"-2\" and it goes through (2,1).\r\n" );
document.write( "So we use the point-slope formula:\r\n" );
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document.write( "\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"\r\n" );
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document.write( "\"y-1=-2%28x-2%29\"\r\n" );
document.write( "\"y-1=-2x%2B4\"\r\n" );
document.write( "\"y=-2x%2B5\"\r\n" );
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document.write( "So we'll draw this perpendicular bisector of the chord:\r\n" );
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document.write( "Where they intersect must be the center of the required\r\n" );
document.write( "circle, so we solve the system of their equations:\r\n" );
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document.write( "\"system%28x%2By=1%2Cy=-2x%2B5%29\"\r\n" );
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document.write( "Solving that system we get (x,y) = (4,-3).\r\n" );
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document.write( "So the center of the circle is (4,-3).\r\n" );
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document.write( "and we can sketch in the circle:\r\n" );
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document.write( "We have its center (4,-3).\r\n" );
document.write( "Now we must find its radius, which is the\r\n" );
document.write( "distance from the center (4,-3) to the point (4,2).\r\n" );
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document.write( "Using the distance formula:\r\n" );
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document.write( "\"d=sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29\"\r\n" );
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document.write( "\"d=sqrt%28%284-4%29%5E2%2B%282-%28-3%29%29%5E2%29\"\r\n" );
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document.write( "\"d=sqrt%28%280%29%5E2%2B%282%2B3%29%29%5E2%29\"\r\n" );
document.write( "\"d=sqrt%280%2B5%5E2%29\"\r\n" );
document.write( "\"d=sqrt%2825%29\"\r\n" );
document.write( "\"d=5\"\r\n" );
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document.write( "So the radius is 5 and the center is (4,-3),\r\n" );
document.write( "so the equation is\r\n" );
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document.write( "\"%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2\"\r\n" );
document.write( "\"%28x-4%29%5E2%2B%28y-%28-3%29%29%5E2=%285%29%5E2\"\r\n" );
document.write( "\"%28x-4%29%5E2%2B%28y%2B3%29%5E2=25\"\r\n" );
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document.write( "Edwin
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