document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1129332>Question 1129332</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a. Show that the point Q(6,2) lies on the circle x^2+y^2-4x+2y-20<BR>\n" );
document.write( "b. Find the equation of tangent to the circle at point Q </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #745907 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=745907\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. Show that the point Q(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) lies on the circle \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2-4x%2B2y-20 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2-4x%2B2y-20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%5E2%2B2%5E2-4%2A6%2B2%2A2-20 ALIGN=MIDDLE  ALT=\"6%5E2%2B2%5E2-4%2A6%2B2%2A2-20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36%2B4-24%2B4-20 ALIGN=MIDDLE  ALT=\"36%2B4-24%2B4-20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=44-44 ALIGN=MIDDLE  ALT=\"44-44\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=>the point Q(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) lies on the circle\r<BR>\n" );
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document.write( "b. Find the equation of tangent to the circle at point Q \r<BR>\n" );
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document.write( "As a tangent is a straight line it is described by an equation in the form <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+-+y%5B1%5D+=+m%28x+-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"+y+-+y%5B1%5D+=+m%28x+-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "You need both a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=point ALIGN=MIDDLE  ALT=\"point\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope ALIGN=MIDDLE  ALT=\"slope\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find its equation.\r<BR>\n" );
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document.write( "You are usually given the point - it's where the tangent meets the circle.\r<BR>\n" );
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document.write( "To find the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope ALIGN=MIDDLE  ALT=\"slope\"   DISABLED_style= \"max-width:90%; height:auto;\" > use the fact that the tangent is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=perpendicular ALIGN=MIDDLE  ALT=\"perpendicular\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=radius ALIGN=MIDDLE  ALT=\"radius\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the point it meets the circle.\r<BR>\n" );
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document.write( "Work out the slope of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=radius ALIGN=MIDDLE  ALT=\"radius\"   DISABLED_style= \"max-width:90%; height:auto;\" > (r) at the point the tangent meets the circle. Then use the equation  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+m_tg+=+-+1%2Fm_r ALIGN=MIDDLE  ALT=\"+m_tg+=+-+1%2Fm_r\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the gradient of the tangent.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-4x%29%2B%28y%5E2%2B2y%29=20 ALIGN=MIDDLE  ALT=\"%28x%5E2-4x%29%2B%28y%5E2%2B2y%29=20\"   DISABLED_style= \"max-width:90%; height:auto;\" >...if you complete squares, you will see that the center of the circle is  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-4x%2Bb%5E2%29-b%5E2%2B%28y%5E2%2B2y%2Bb%5E2%29-b%5E2=20 ALIGN=MIDDLE  ALT=\"%28x%5E2-4x%2Bb%5E2%29-b%5E2%2B%28y%5E2%2B2y%2Bb%5E2%29-b%5E2=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-4x%2B2%5E2%29-2%5E2%2B%28y%5E2%2B2y%2B1%5E2%29-b%5E2=20 ALIGN=MIDDLE  ALT=\"%28x%5E2-4x%2B2%5E2%29-2%5E2%2B%28y%5E2%2B2y%2B1%5E2%29-b%5E2=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%29-4%2B%28y%2B1%29%5E2%29-1=20 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%29-4%2B%28y%2B1%29%5E2%29-1=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%29%2B%28y%2B1%29%5E2%29=25 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%29%2B%28y%2B1%29%5E2%29=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Q(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)=>\r<BR>\n" );
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document.write( " the slope of the radius \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m_r=%28-1-2%29%2F%282-6%29=-3%2F-4=3%2F4 ALIGN=MIDDLE  ALT=\"m_r=%28-1-2%29%2F%282-6%29=-3%2F-4=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the slope of the tangent is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m_tg=-1%2F%283%2F4%29=-4%2F3 ALIGN=MIDDLE  ALT=\"m_tg=-1%2F%283%2F4%29=-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and equation af the tangent is:\r<BR>\n" );
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document.write( "Q(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m_tg=-4%2F3 ALIGN=MIDDLE  ALT=\"m_tg=-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+2+=+-%284%2F3%29%28x+-6%29 ALIGN=MIDDLE  ALT=\"y+-+2+=+-%284%2F3%29%28x+-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y++=+-%284%2F3%29x+-6%28-4%2F3%29%2B2 ALIGN=MIDDLE  ALT=\"y++=+-%284%2F3%29x+-6%28-4%2F3%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y++=+-%284%2F3%29x+%2B8%2B2 ALIGN=MIDDLE  ALT=\"y++=+-%284%2F3%29x+%2B8%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y++=+-%284%2F3%29x+%2B10 ALIGN=MIDDLE  ALT=\"y++=+-%284%2F3%29x+%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28+600%2C+600%2C+-15%2C+15%2C+-15%2C+15%2C%0D%0Acircle%286%2C2%2C.12%29%2Clocate%286%2C2%2CQ%286%2C2%29%29%2Ccircle%282%2C-1%2C.12%29%2Clocate%282%2C-1%2CC%282%2C-1%29%29%2C%0D%0A+graph%28+600%2C+600%2C+-15%2C+15%2C+-15%2C+15%2C-%284%2F3%29x+%2B10%2Csqrt%2825-%28x-2%29%5E2%29-1%2C++-sqrt%2825-%28x-2%29%5E2%29-1%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );