document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1078608>Question 1078608</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the circle that passes through (1, 2) and (3,4); tangent to 4x - 3y - 2 = 0 at (-1, -2). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #693268 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=693268\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Strange problem.<BR>\n" );
document.write( "At first sight, there seems to be too much information.<BR>\n" );
document.write( "Excess information may be useless, but consistent with the rest.<BR>\n" );
document.write( "In this case, there is contradictory information.<BR>\n" );
document.write( "As written, there is no solution.<BR>\n" );
document.write( "Maybe there is a typo.<BR>\n" );
document.write( "Maybe the circle is supposed to just be tangent to the line at (-1,-2),<BR>\n" );
document.write( "and also pass through one other given point.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If the circle is tangent to any line at (-1,-2),<BR>\n" );
document.write( "the circle passes through (-1,-2).<BR>\n" );
document.write( "Three non-colinear points determine a circle.<BR>\n" );
document.write( "In other words, given three points that are not on the same line,<BR>\n" );
document.write( "there is one and only one circle that passes through all three.<BR>\n" );
document.write( "So, there is only one circle that passes through<BR>\n" );
document.write( "(1,2) , (3,4) and (-1,-2),<BR>\n" );
document.write( "and that circle is not tangent to the line indicated.<BR>\n" );
document.write( "That circle, with the tree points indicated,<BR>\n" );
document.write( "and the line 4x-3y-2=0 is shown below.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C22%2C-18%2C8%2Cgrid%281%29%2C%0D%0Acircle%281%2C2%2C0.3%29%2Ccircle%28-1%2C-2%2C0.3%29%2C%0D%0Acircle%283%2C4%2C0.3%29%2Cgreen%28line%28-4%2C-6%2C8%2C10%29%29%2C%0D%0Ared%28circle%2810%2C-5%2Csqrt%28130%29%29%29%2Cred%28circle%2810%2C-5%2C0.3%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "I can make a circle go through (1,2) and (3,4) ,<BR>\n" );
document.write( "and be tangent to the line 4x-3y-2=0,<BR>\n" );
document.write( "but I cannot choose the point of tangency,<BR>\n" );
document.write( "and it is not (-1,-2).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C6%2C-1%2C9%2Cgrid%281%29%2C%0D%0Acircle%281%2C2%2C0.1%29%2Ccircle%282.6%2C2.8%2C0.1%29%2C%0D%0Acircle%283%2C4%2C0.1%29%2Cgreen%28line%28-4%2C-6%2C8%2C10%29%29%2C%0D%0Ared%28circle%281%2C4%2C2%29%29%2Cred%28circle%281%2C4%2C0.1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE: There are a few things to know to find the equation of a circle given some points.<BR>\n" );
document.write( "1) If you know two points on the circle,<BR>\n" );
document.write( "the center of the circle is on the perpendicular bisector<BR>\n" );
document.write( "of the line joining those two points.<BR>\n" );
document.write( "2) If you are given a tangent and the point of tangency,<BR>\n" );
document.write( "the center of the circle is on a line<BR>\n" );
document.write( "perpendicular to the tangent<BR>\n" );
document.write( "and passing through the point of tangency.<BR>\n" );
document.write( "3) If a line through two points A and B on the circle,<BR>\n" );
document.write( "and a tangent to the circle intersect at a point P,<BR>\n" );
document.write( "the point of tangency, T, is located so that <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PT%5E2=PA%2APB ALIGN=MIDDLE  ALT=\"PT%5E2=PA%2APB\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "4) with the center of the circle and a point on the circle,<BR>\n" );
document.write( "you can find the radius (the distance between those points),<BR>\n" );
document.write( "and have all the information needed to write the equation of the circle. </SPAN>\n" );
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