document.write( "Question 1087524: Write the equations of the circles satisfying the condition, tangent to 5x - y - 17 = 0 at (4,3) and also tangent to x - 5y - 5 = 0 \n" ); document.write( "
Algebra.Com's Answer #701894 by ikleyn(52915)\"\" \"About 
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\n" ); document.write( "Write the equations of the circles satisfying the condition, tangent to 5x - y - 17 = 0 at (4,3) and also tangent to x - 5y - 5 = 0
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document.write( "The key step to solve this problem is to find the center of the circle.\r\n" );
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document.write( "From one side, the center lies on the perpendicular to the first line passing through the given point.\r\n" );
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document.write( "From the other side, the center lies on the angle bisector of the angle formed by the two given lines \r\n" );
document.write( "(since this angle bisector is the locus of points equidistant from the lines - very elementary basic property of the angle bisector . . . )\r\n" );
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document.write( "So, our goal is to find the intersection point of the perpendicular and the angle bisector.\r\n" );
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document.write( "Since we have, actually, TWO different adjacent angles between the given lines, we will have TWO different angle bisectors that would create \r\n" );
document.write( "two different intersection points and, correspondingly, two different circles.\r\n" );
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document.write( "OK. So we just have an idea on what we want and how to do it.  Now let's implement it.\r\n" );
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document.write( "The straight line 5x - y - 17 = 0 has the slope m = 5.  Hence, the perpendicular line has the slope \"-1%2F5\" = -0.2.\r\n" );
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document.write( "The equation of the line having the slope -0.2 and passing through the point (4,3) is\r\n" );
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document.write( "y - 3 = -0.2*(x-4),   or, which is the same,   x + 5y = 19.  (1)\r\n" );
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document.write( "Thus the perpendicular to the first given line at the given point is (1).\r\n" );
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document.write( "    So, the first part of the work is done.   We got the equation of the perpendicular line.\r\n" );
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document.write( "    Next goal is to write an equation (equations) for the angle bisector/bisectors to the two given lines.\r\n" );
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document.write( "Probably, you never did it at school, but I will teach you right now on how to do it.\r\n" );
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document.write( "The distance from the point (x,y) in a coordinate plane to the line  5x - y - 17 = 0   is  \"abs%285x+-+y+-17%29%2Fsqrt%285%5E2+%2B+%28-1%29%5E2%29\" = \"abs%285x+-+y+-17%29%2Fsqrt%2826%29\".\r\n" );
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document.write( "    See the lesson  The distance from a point to a straight line in a coordinate plane in this site.\r\n" );
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document.write( "Similarly, the distance from the point (x,y) in a coordinate plane to the line  x - 5y - 5 = 0  is  \"abs%28x+-5y+-5%29%2Fsqrt%281%5E2+%2B+%28-5%29%5E2%29\" = \"abs%28x+-+5y+-5%29%2Fsqrt%2826%29\".\r\n" );
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document.write( "If the point (x,y) lies on the angle bisector between the given lines, then the point (x,y) is equidistant from the lines,\r\n" );
document.write( "which gives you an equation \r\n" );
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document.write( "    \"abs%285x+-+y+-17%29%2Fsqrt%2826%29\" = \"abs%28x+-+5y+-5%29%2Fsqrt%2826%29\",  or, canceling \"sqrt%2826%29\" in both sides, equivalently,\r\n" );
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document.write( "    \"abs%285x+-+y+-17%29\" = \"abs%28x+-+5y+-5%29\".    (2)\r\n" );
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document.write( "From learning absolute values in the school, you must know that the equation (2) is equivalent to the SET of these two equations\r\n" );
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document.write( "    5x - y - 17 =   x - 5y -5,       (3)    and\r\n" );
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document.write( "    5x - y - 17 = -(x - 5y -5).      (4)\r\n" );
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document.write( "Next, simplifying, you have \r\n" );
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document.write( "    4x + 4y = 12                     (5)     instead of (3),   and\r\n" );
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document.write( "    6x - 6y = 22                     (6)     instead of (4).\r\n" );
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document.write( "Actually, equation (5) is for one angle bisector, while equation (6) is for another angle bisector.\r\n" );
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document.write( "    By the way, notice in your mind that the straight lines (5) and (6) are perpendicular, \r\n" );
document.write( "    which you may expect for angle bisectors of the two adjacent supplementary angles. \r\n" );
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document.write( "Now we are at the finish line.  We must solve the equation (1) with the equation (5) to get one intersection point (=the center),\r\n" );
document.write( "and we must solve the equation (1) with the equation (6) to get the second intersection point (=the center of the second circle).\r\n" );
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document.write( "Solving equations (1) and (5) together\r\n" );
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document.write( " x + 5y = 19,\r\n" );
document.write( "4x + 4y = 12,\r\n" );
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document.write( "you get the solution x = -1, y = 4.   It is your first center.\r\n" );
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document.write( "Solving equations (1) and (6) together\r\n" );
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document.write( " x + 5y = 19,\r\n" );
document.write( "6x - 6y = 22,\r\n" );
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document.write( "you get the solution x = \"56%2F9\", y = \"23%2F9\".   It is your second center (of the second circle).\r\n" );
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document.write( "Having the centers, you can easily calculate the radii of the circles as distances to the given point.\r\n" );
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document.write( "Since Edwin just gave the plot in his post, I will not repeat it again.\r\n" );
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