document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=993749>Question 993749</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find point P(x,y) on line y=2x-1 so that its distance from point Q(4,3) is 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #613019 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=613019\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can solve this graphically easy enough if you have good graphing software.<BR>\n" );
document.write( "using one, i found that two points satisfy the requirements.<BR>\n" );
document.write( "those points are (2,3) and (2.8,4.60)\r<BR>\n" );
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document.write( "what you are looking for on the graph is the intersection of the line y = 2x-1 with the intersection of the circle whose equation is (x-4)^2 + (y-3)^2 = 4.\r<BR>\n" );
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document.write( "without graphing software it's not quite as easy, but can be done.\r<BR>\n" );
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document.write( "you are still looking for the intersection of the graph of the equations shown above, but you have to do it algebraically.\r<BR>\n" );
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document.write( "you have 2 equations:\r<BR>\n" );
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document.write( "y = 2x - 1<BR>\n" );
document.write( "(x-4)^2 + (y-3)^2 = 4\r<BR>\n" );
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document.write( "the equation of y = 2x - 1 is a straight line.\r<BR>\n" );
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document.write( "the equation of (x-4)^2 + (y-3)^2 = 4 is a circle whose center is (4,3) and whose radius is 2.\r<BR>\n" );
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document.write( "replace y in the second equation with the value of y from the first equation to get:\r<BR>\n" );
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document.write( "(x-4)^2 + (2x-1-3)^2 = 4\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "(x-4)^2 + (2x-4)^2 = 4\r<BR>\n" );
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document.write( "simplify the equation to get:\r<BR>\n" );
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document.write( "x^2 - 8x + 16 + 4x^2 - 16x + 16 = 4\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "5x^2 - 24x + 32 = 4\r<BR>\n" );
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document.write( "subtract 4 from both sides of the equation to get:\r<BR>\n" );
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document.write( "5x^2 - 24x + 28 = 0\r<BR>\n" );
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document.write( "use the quadratic formula to find the values of x that satisfy that equation.\r<BR>\n" );
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document.write( "you will find that the values of x = 2 and x = 2.8 satisfy both equations and are therefore intersections of both equations on the graph.\r<BR>\n" );
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document.write( "when x = 2, y = 2x-1 becomes y = 3.<BR>\n" );
document.write( "when x = 2.8, y = 2x-1 becomes 4.6.\r<BR>\n" );
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document.write( "the points on the circle that are also on the line are therefore (2,3) and (2.8,4.6)\r<BR>\n" );
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document.write( "both these points should be 2 units away from the center of the circle because the line segments formed by each of these points and the center of the circle is a radius of the circle.\r<BR>\n" );
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document.write( "the distance between two points on the graph is given by the equation:\r<BR>\n" );
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document.write( "d = sqrt((x1-x2)^2 + (y1-y2)^2).\r<BR>\n" );
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document.write( "the distance between (2,3) and (4,3) is equal to sqrt((4-2)^2 + (3-3)^2) which is equal to sqrt(2^2) which is equal to sqrt(4) which is equal to 2.\r<BR>\n" );
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document.write( "the distance between (2.8,4.6) and (4,3) is equal to sqrt((4-2.8)^2 + (3-4.6)^2) which is equal to sqrt(1.2)^2 + (-1.6)^2) which is equal to sqrt(1.44 + 2.56) which is equal to sqrt(4) which is equal to 2.\r<BR>\n" );
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document.write( "either one of these points will satisfy the requirements of this problem.\r<BR>\n" );
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document.write( "the graph of the line and the circle are shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/100201.jpg\" alt=\"%%%\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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