document.write( "Question 832263: Find the General Equation of the circle\r
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Algebra.Com's Answer #502023 by math-vortex(648) $\"\"$ $\"About$
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document.write( "Hi, again--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "Find the general equation of the circle with its center at the origin and tangent to line 5x + 12y = 26.\r\n" );
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document.write( "A SOLUTION:\r\n" );
document.write( "We want the equation for the circle in this form.\r\n" );
document.write( "x^2 + y^2 + Dx + Ey + F = 0\r\n" );
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document.write( "Our first step is to find the point of tangency for the line 5x + 12y = 26. We use the fact that line through \r\n" );
document.write( "the center of the circle and the point of tangency (called the normal line) and the tangent line are \r\n" );
document.write( "perpendicular. \r\n" );
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document.write( "Find the slope of the tangent line. (Convert the equation to slope-intercept form.)\r\n" );
document.write( "5x + 12y = 26\r\n" );
document.write( "12y = -5x + 26\r\n" );
document.write( "y = -(5/12)x + 26/12\r\n" );
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document.write( "The slope of the tangent line is -5/12.\r\n" );
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document.write( "Find the slope of the normal line. Recall that the slopes of perpendicular lines are related by \r\n" );
document.write( ". The slope of the normal line is 12/5 since (-5/12)(12/5) = -1.\r\n" );
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document.write( "Use the point-slope form to find an equation for the normal line. The slope is 12/5 the point is (0,0) \r\n" );
document.write( "since the line passes through the origin.\r\n" );
document.write( "y - 0 = (12/5)(x-0)\r\n" );
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document.write( "Simplify  to give the equation for the normal line.\r\n" );
document.write( "y = (12/5)x\r\n" );
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document.write( "Since the normal line and tangent line intersect at the point of tangency, we solve the system of equations \r\n" );
document.write( "to find that intersection point.\r\n" );
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document.write( "5x + 12y = 26\r\n" );
document.write( "y = (12/5)x\r\n" );
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document.write( "Substitute (12/5)x for y in the first equation.\r\n" );
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document.write( "5x + 12(12/5)x = 26\r\n" );
document.write( "5x + 144x/5 = 26\r\n" );
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document.write( "Multiply every term by 5 to clear the denominator.\r\n" );
document.write( "25x + 144x = 130\r\n" );
document.write( "169x = 130\r\n" );
document.write( "x = 130/169\r\n" );
document.write( "x = 10/13\r\n" );
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document.write( "Substitute 10/13 for x in the second equation.\r\n" );
document.write( "y = (12/5)x\r\n" );
document.write( "y = (12/5)(10/13)\r\n" );
document.write( "y = 24/13\r\n" );
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document.write( "The normal line and the tangent line intersect at the point (10/13, 24/13). More important, this point is \r\n" );
document.write( "on the circle. The distance between this point and origin is the radius of the circle. Find the radius r \r\n" );
document.write( "using the distance formula.\r\n" );
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document.write( "An equation for a circle with radius r and center (h,k) is\r\n" );
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document.write( "Substitute 0 for h, 0 for k and 26/13 for r.\r\n" );
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document.write( "Simplify and rewrite the equation in general form.\r\n" );
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document.write( "Hope this helps! Feel free to email if you have any questions about the solution.\r\n" );
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document.write( "Good luck with your math,\r\n" );
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document.write( "Mrs. F\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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