document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=913984>Question 913984</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem:\r<BR>\n" );
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document.write( "A circle of radius 1 is centered at the origin and passes through the point (1/2,3&#8730;/2).<BR>\n" );
document.write( "(a) Find an equation for the line through the origin and the point (1/2,3&#8730;/2). <BR>\n" );
document.write( "y= \r<BR>\n" );
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document.write( "(b) Find an equation for the tangent line to the circle at (1/2,3&#8730;/2). <BR>\n" );
document.write( "y=  <BR>\n" );
document.write( "[Hint: A tangent line is perpendicular to the radius at the point of tangency.] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #554837 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=554837\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A circle of radius and centered at the origin and passes through the point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2F2%2Csqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"P%281%2F2%2Csqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "The equation for a line that starts with that starts with \"y =\" is the equation in slope-intercept form,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=m%2Ax%2Bb ALIGN=MIDDLE  ALT=\"y=m%2Ax%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the y-intercept.<BR>\n" );
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document.write( "(a) The line through the origin, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O%280%2C0%29 ALIGN=MIDDLE  ALT=\"O%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2F2%2Csqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"P%281%2F2%2Csqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has a y-intercept of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=0 ALIGN=MIDDLE  ALT=\"b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <BR>\n" );
document.write( "a slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28sqrt%283%29%2F2-0%29%2F%281%2F2-0%29=%28sqrt%283%29%2F2%29%2F%281%2F2%29=%28sqrt%283%29%2F2%29%2A%282%2F1%29=sqrt%283%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The equation for the line hrough the origin and the point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2F2%2Csqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"P%281%2F2%2Csqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > that starts with \"y =\" is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=+sqrt%283%29x%29 ALIGN=MIDDLE  ALT=\"highlight%28y=+sqrt%283%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "(b) The tangent line to the circle at the point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2F2%2Csqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"P%281%2F2%2Csqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > goes through the point of tangency, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2F2%2Csqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"P%281%2F2%2Csqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The radius of the circle at the point of tangency is the line segment that connects the points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O%280%2C0%29 ALIGN=MIDDLE  ALT=\"O%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2F2%2Csqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"P%281%2F2%2Csqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That line segment is part of the line with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+sqrt%283%29x ALIGN=MIDDLE  ALT=\"y=+sqrt%283%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=sqrt%283%29 ALIGN=MIDDLE  ALT=\"m=sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > found in part a).<BR>\n" );
document.write( "According to the hint, the tangent you are looking for is perpendicular to that line and passes through the point of tangency, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2F2%2Csqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"P%281%2F2%2Csqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The line perpendicular to a line with slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >, has a slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Fm ALIGN=MIDDLE  ALT=\"-1%2Fm\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, for the tangent you are looking for,<BR>\n" );
document.write( "the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Fsqrt%283%29 ALIGN=MIDDLE  ALT=\"-1%2Fsqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is more elegantly expressed as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2Fsqrt%283%29%29%2A%28sqrt%283%29%2Fsqrt%283%29%29=-sqrt%283%29%2F3 ALIGN=MIDDLE  ALT=\"%28-1%2Fsqrt%283%29%29%2A%28sqrt%283%29%2Fsqrt%283%29%29=-sqrt%283%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The equation of that tangent in point-slope form, based on point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2F2%2Csqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"P%281%2F2%2Csqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y-sqrt%283%29%2F2=%28-sqrt%283%29%2F3%29%28x-1%2F2%29%29 ALIGN=MIDDLE  ALT=\"highlight%28y-sqrt%283%29%2F2=%28-sqrt%283%29%2F3%29%28x-1%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you wanted to express that equation in slope intercept form, you would \"distribute\" and \"solve for y\", like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-sqrt%283%29%2F2=%28-sqrt%283%29%2F3%29%28x-1%2F2%29 ALIGN=MIDDLE  ALT=\"y-sqrt%283%29%2F2=%28-sqrt%283%29%2F3%29%28x-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-sqrt%283%29%2F2=%28-sqrt%283%29%2F3%29x-%281%2F2%29%28-sqrt%283%29%2F3%29 ALIGN=MIDDLE  ALT=\"y-sqrt%283%29%2F2=%28-sqrt%283%29%2F3%29x-%281%2F2%29%28-sqrt%283%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-sqrt%283%29%2F2=%28-sqrt%283%29%2F3%29x%2Bsqrt%283%29%2F6 ALIGN=MIDDLE  ALT=\"y-sqrt%283%29%2F2=%28-sqrt%283%29%2F3%29x%2Bsqrt%283%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-sqrt%283%29%2F3%29x%2Bsqrt%283%29%2F6%2Bsqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"y=%28-sqrt%283%29%2F3%29x%2Bsqrt%283%29%2F6%2Bsqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-sqrt%283%29%2F3%29x%2Bsqrt%283%29%2F6%2B3sqrt%283%29%2F6 ALIGN=MIDDLE  ALT=\"y=%28-sqrt%283%29%2F3%29x%2Bsqrt%283%29%2F6%2B3sqrt%283%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-sqrt%283%29%2F3%29x%2B4sqrt%283%29%2F6 ALIGN=MIDDLE  ALT=\"y=%28-sqrt%283%29%2F3%29x%2B4sqrt%283%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=%28-sqrt%283%29%2F3%29x%2B2sqrt%283%29%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28y=%28-sqrt%283%29%2F3%29x%2B2sqrt%283%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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