document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=190519>Question 190519</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0395977258/algebrahomeworkh>Algebra and Trigonometry Structure and Method book 2</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> I have been struggling on this math problem and I was wondering if someone could help me? Please and Thank You!! I would deeply appreciate!!\r<BR>\n" );
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document.write( "Circles<BR>\n" );
document.write( "Find an equation of the circle described.<BR>\n" );
document.write( "Center in quadrant four; tangent to the lines x=1, x=9 and y=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142971 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=142971\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Just a quick reminder: the equation of any circle is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where (h,k) is the center and \"r\" is the radius.\r<BR>\n" );
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document.write( "If you draw the lines x=1 and x=9, you'll get two vertical lines at x=1 and x=9. Since the circle is tangent to both these lines, and the two lines are parallel, this means that endpoints of the diameter touches these lines. So the endpoints of the diameter are the points (1,y) and (9,y) where \"y\" is some unknown number\r<BR>\n" );
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document.write( "So this basically means that the endpoints of the diameter have x-coordinates of 1 and 9. Now let's find the midpoint of the diameter to get the center of the circle:\r<BR>\n" );
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document.write( "First, add 1 to 9 to get 10. Now take half of 10 to get 5. \r<BR>\n" );
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document.write( "So the midpoint of the diameter has an x-coordinate of 5. This means that the x-coordinate of the center is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > which also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=5 ALIGN=MIDDLE  ALT=\"h=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Take note that the distance from 5 to 9 is 4 and the distance from 1 to 5 is 4. So this means that the radius of the circle is 4 units. So we've found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=4 ALIGN=MIDDLE  ALT=\"r=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now because the circle is also tangent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the center is also 4 units away from points of the form (x,0). The only two points that have x-coordinates of 5 and are 4 units away from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are (5,-4) and (5,4). But we have to remember that the problem states that: \"Center in quadrant four\". So this eliminates the second point leaving the only possible center of (5,-4)\r<BR>\n" );
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document.write( "So the center of the circle is (5,-4) and the radius is 4 units. This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=5 ALIGN=MIDDLE  ALT=\"h=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-4 ALIGN=MIDDLE  ALT=\"k=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=4 ALIGN=MIDDLE  ALT=\"r=4\"   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-h%29%5E2%2B%28y-k%29%5E2=r%5E2 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%5E2%2B%28y-%28-4%29%29%5E2=4%5E2 ALIGN=MIDDLE  ALT=\"%28x-5%29%5E2%2B%28y-%28-4%29%29%5E2=4%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=5 ALIGN=MIDDLE  ALT=\"h=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-4 ALIGN=MIDDLE  ALT=\"k=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=4 ALIGN=MIDDLE  ALT=\"r=4\"   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-5%29%5E2%2B%28y%2B4%29%5E2=4%5E2 ALIGN=MIDDLE  ALT=\"%28x-5%29%5E2%2B%28y%2B4%29%5E2=4%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%5E2%2B%28y%2B4%29%5E2=16 ALIGN=MIDDLE  ALT=\"%28x-5%29%5E2%2B%28y%2B4%29%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 4 to get 16\r<BR>\n" );
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document.write( "So the equation of the circle with the given properties is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%5E2%2B%28y%2B4%29%5E2=16 ALIGN=MIDDLE  ALT=\"%28x-5%29%5E2%2B%28y%2B4%29%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's a drawing to confirm\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+0%29%2C+%0D%0Ared%28circle%285%2C-4%2C4%29%29%2C%0D%0Ablue%28line%281%2C-20%2C1%2C20%29%29%2C%0D%0Ablue%28line%289%2C-20%2C9%2C20%29%29%2C%0D%0Ablue%28line%28-20%2C0%2C20%2C0%29%29%0D%0Acircle%285%2C-4%2C0.03%29%2Ccircle%285%2C-4%2C0.05%29%2Ccircle%285%2C-4%2C0.08%29%2Ccircle%285%2C-4%2C0.10%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%5E2%2B%28y%2B4%29%5E2=16 ALIGN=MIDDLE  ALT=\"%28x-5%29%5E2%2B%28y%2B4%29%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with center (5,-4)) with the blue tangent lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (vertical lines) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (blue line overlapping the x-axis) </SPAN>\n" );
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