document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158612>Question 1158612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the equation of the circle that passes through the 3 points (5, -7), (2, -4), and (5, -3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #781559 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=781559\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The key to this problem is the fact that the perpendicular bisector of any chord of a circle will pass through the center of the circle.  Three non-colinear points, such as the three points you are given are the endpoints of three different chords of the desired circle.  You only need to consider two of those chords.\r<BR>\n" );
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document.write( "Step 1:  Choose any two of the three given points and use the midpoint formulas to find the coordinates of the chord defined by the two given endpoints. Hint: I would choose (5,-7) and (5,-3) as one of the pairs since these two points lie on a vertical line and this will greatly simplify your calculations going forward.\r<BR>\n" );
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document.write( "Step 2:  Calculate the slope of the line that contains the two chosen points. Remember that a vertical line has an undefined slope and an equation of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\"> some constant value.\r<BR>\n" );
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document.write( "Step 3:  Calculate the negative reciprocal of the value you obtained in step 2.  This is the slope of a perpendicular to the chord.  A perpendicular to a vertical line is a horizontal line that has a slope of zero. I.e., an equation of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\"> some constant value.\r<BR>\n" );
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document.write( "Step 4:  Use the Point-Slope form of an equation of a straight line with the midpoint coordinates and the slope of the perpendicular found in steps 1 and 3 to write an equation of the perpendicular bisector of your first chord.\r<BR>\n" );
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document.write( "Step 5:  Repeat Steps 1 through 4 using a different pair of the given points.  You will then have equations of two perpendicular bisectors of two chords of the desired circle, which, as was noted earlier, intersect at the center of the circle.\r<BR>\n" );
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document.write( "Step 6:  Use any convenient method to solve the 2X2 system of equations that resulted from steps 4 and 5.  The unique solution will be the coordinates of the center of the desired circle.\r<BR>\n" );
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document.write( "Step 7:  Use the distance formula to calculate the distance from the center to any one of the given points.  This will be the radius of the desired circle. Hint: Don't actually take the square root as the final step of calculating the measure of the radius because your final result requires the radius squared.\r<BR>\n" );
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document.write( "Step 8:  Write the equation of the circle:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ h)^2\ +\ (y\ -\ k)^2\ =\ r^2\">\r<BR>\n" );
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document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h\"> is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the circle center, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of the circle center, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r\"> is the radius of the circle.  All three values being the results of your Step 6 and Step 7 calculations.  Be careful with your signs -- if either coordinate is negative, then you retain that sign and end up with a plus (minus a minus) in the final expression.\r<BR>\n" );
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document.write( "Enjoy.<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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