document.write( "Question 1158991: Find an equation for the line that goes through the two intersection points of the circle
\n" ); document.write( "x2 +y2 =25andthecircle(x−8)2 +(y−4)2 =65. \n" ); document.write( "

Algebra.Com's Answer #782024 by ikleyn(52810) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            Notice that the problem DOES NOT ASK YOU to find the coordinates of the intersection points.\r
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\n" ); document.write( "\n" ); document.write( "            It only ask you to find a linear equation for the line, which goes through these points.\r
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\n" ); document.write( "\n" ); document.write( "            Therefore, MUCH SIMPLER solution is possible.\r
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document.write( "If the point (x,y) lies on both curves/(circles), given by their equations\r\n" );
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document.write( "    x^2 + y^2 = 25,            (1)\r\n" );
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document.write( "    (x-8)^2 + (y-4)^2 = 65,    (2)\r\n" );
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document.write( "then this point belongs to the intersection of these curves and satisfies to the equation \r\n" );
document.write( "which is the difference of equations (1) and (2).\r\n" );
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document.write( "The difference of these equations CANCEL the quadratic terms, leaving only linear terms\r\n" );
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document.write( "    x^2            + y^2           = 25     (1)\r\n" );
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document.write( "    x^2 - 16x + 64 + y^2 - 8y + 16 = 65     (2)\r\n" );
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document.write( "---------------------------------------------------  Take the difference eq(1) - eq(2)\r\n" );
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document.write( "          16x - 64       + 8y  - 16 = 25 - 65\r\n" );
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document.write( "          16x + 8y                  = 25 - 65 + 64 + 16\r\n" );
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document.write( "          16x + 8y                  = 40\r\n" );
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document.write( "           2x + y                   = 5.\r\n" );
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document.write( "ANSWER.  The equation, the problem asks for, is  2x + y = 5.\r\n" );
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