document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=10806>Question 10806</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear Sir/Madam,\r<BR>\n" );
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document.write( "I am having difficulty finding the equation to the following problem. \r<BR>\n" );
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document.write( "The center of a circle is (1,2). A line touches the circle at a point (1,4). The line is tangent to a circle at that point. Find the equation of the line. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #5673 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=5673\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Try this:\r<BR>\n" );
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document.write( "If the center of the circle is at (1, 2) and a line is tangent to the circle at point (1, 4), then:\r<BR>\n" );
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document.write( "a) The radius of the circle is 2 and,<BR>\n" );
document.write( "b) The equation of the line is y = 4.\r<BR>\n" );
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document.write( "The point of tangency has the same x-coordinate as the center of the circle, this means that the line has zero slope.\r<BR>\n" );
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document.write( "A line that has zero slope and passes through the point (1, 4) has the a y-intercept of (0, 4) and the equation is:\r<BR>\n" );
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document.write( "y = (0)x + 4 or y = 4\r<BR>\n" );
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document.write( "I had some additional thoughts on this problem.  While they won't change my solution, I think they might add a little to the explanation.\r<BR>\n" );
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document.write( "Consider this:<BR>\n" );
document.write( "The line segment from the center of the circle (1, 2) to the point of tangency (1, 4) is the radius of the circle and has a slope of:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y2-y2%29%2F%28x2-x1%29 ALIGN=MIDDLE  ALT=\"m+=+%28y2-y2%29%2F%28x2-x1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%284-2%29%2F%281-1%29 ALIGN=MIDDLE  ALT=\"m+=+%284-2%29%2F%281-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+2%2F0 ALIGN=MIDDLE  ALT=\"m+=+2%2F0\"   DISABLED_style= \"max-width:90%; height:auto;\" > This means, of course, the this line segment has an undefined slope, i.e., it's vertical.\r<BR>\n" );
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document.write( "Now the line is tangent to the circle at the point (1, 4) and, as you know, lines that are tangent to a circle at a given point are perpendicular to the radius at that point. And lines that are perpendicular have slopes that are the negative reciprocal of each other.<BR>\n" );
document.write( "The negative reciprocal of 4/0 is -0/4 or just 0.\r<BR>\n" );
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document.write( "Therefore, the line that is tangent to the circle in the problem has a slope of zero and it passes through the point (1, 4), its equation is: y = 4 <BR>\n" );
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