document.write( "Question 1165331: Find the equation of a circle tangent to the line 3x-4y-16=0 and containing the points (1,8) and (-2,-1). \n" ); document.write( "

Algebra.Com's Answer #789835 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "There is surely a better way to find the answer than this....

\n" ); document.write( "But it works, so here it is.

\n" ); document.write( "Perhaps another tutor will find a simpler but more elegant path to the answer and show it to us.

\n" ); document.write( "The calculations required in solving the problem by this method are ugly. I will not show the detailed calculations -- only the results. If you want to learn something from this problem, you will go through the calculations in detail by yourself.

\n" ); document.write( "Let the center of the circle be O(h,k).

\n" ); document.write( "(1) The center of the circle is on the perpendicular bisector of the segment defined by the two given points, A(1,8) and B(-2,-1).

\n" ); document.write( "-- find the slope and midpoint of segment AB
\n" ); document.write( "-- find the equation of the perpendicular bisector of AB

\n" ); document.write( "That equation is $\"y+=+%28-1%2F3%29x%2B%2810%2F3%29\"$

\n" ); document.write( "Since (h,k) is on that perpendicular bisector, we know $\"k+=+%28-1%2F3%29h%2B%2810%2F3%29\"$ .

\n" ); document.write( "(2) Since the circle is tangent to the line 3x-4y-16=0, use the point-to-line distance formula to find that the distance from (h,k) to that line is

\n" ); document.write( " $\"abs%283h-4k-16%29%2Fsqrt%283%5E2%2B4%5E2%29+=+abs%283h-4k-16%29%2F5\"$

\n" ); document.write( "(3) The distances from (h,k) to the line and from (h,k) to either of the given points are the same. Use $\"k+=+%28-1%2F3%29h-%2810%2F3%29\"$ (from step (1) above) in the formulas for those two distances to get an equation to solve for h, the x-coordinate of the center of the circle.

\n" ); document.write( "(The calculations there are REALLY ugly -- use a graphing calculator to solve the equation that says those distances are equal.)

\n" ); document.write( "The result of all that ugly calculation is that h=1; that makes k=3.

\n" ); document.write( "So the center of the circle is (1,3).

\n" ); document.write( "It is then a simple matter (thankfully!) to show that the distances from (1,3) to each of the given points and to the given line are all 5.

\n" ); document.write( "And then, finally, the equation of the circle is

\n" ); document.write( " $\"%28x-1%29%5E2%2B%28y-3%29%5E2+=+25\"$

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