document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1068273>Question 1068273</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an equation(s) of the circle(s) of radius 4 with center on the line 4x + 3y + 7 = 0 and tangent to 3x + 4y + 34 = 0<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #683504 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=683504\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the line perpendicular to the tangent line that goes through the circle center. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%2B34=0 ALIGN=MIDDLE  ALT=\"3x%2B4y%2B34=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y=-3x-34 ALIGN=MIDDLE  ALT=\"4y=-3x-34\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2F4%29x-17%2F2 ALIGN=MIDDLE  ALT=\"y=-%283%2F4%29x-17%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Perpendicular lines have slopes that are negative reciprocals.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283%2F4%29m%5B2%5D=-1 ALIGN=MIDDLE  ALT=\"-%283%2F4%29m%5B2%5D=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B2%5D=4%2F3 ALIGN=MIDDLE  ALT=\"m%5B2%5D=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let the circle center be located at (h,k) and let the intersection point of the tangent line and the perpendicular line be (a,b).<BR>\n" );
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document.write( "So you know that,<BR>\n" );
document.write( "1.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%2B4b%2B34=0 ALIGN=MIDDLE  ALT=\"3a%2B4b%2B34=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4h%2B3k%2B7=0 ALIGN=MIDDLE  ALT=\"4h%2B3k%2B7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perpendicular line goes through both (h,k) and (a,b) so,<BR>\n" );
document.write( "3.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k-b=%284%2F3%29%28h-a%29 ALIGN=MIDDLE  ALT=\"k-b=%284%2F3%29%28h-a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and from the distance formula,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2=%28h-a%29%5E2%2B%28k-b%29%5E2 ALIGN=MIDDLE  ALT=\"4%5E2=%28h-a%29%5E2%2B%28k-b%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "4.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h-a%29%5E2%2B%28k-b%29%5E2=16 ALIGN=MIDDLE  ALT=\"%28h-a%29%5E2%2B%28k-b%29%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute 3 into 4,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h-a%29%5E2%2B%2816%2F9%29%28h-a%29%5E2=16 ALIGN=MIDDLE  ALT=\"%28h-a%29%5E2%2B%2816%2F9%29%28h-a%29%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825%2F9%29%28h-a%29%5E2=16 ALIGN=MIDDLE  ALT=\"%2825%2F9%29%28h-a%29%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h-a%29%5E2=%2816%2A9%29%2F25 ALIGN=MIDDLE  ALT=\"%28h-a%29%5E2=%2816%2A9%29%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h-a=+0+%2B-+12%2F5 ALIGN=MIDDLE  ALT=\"h-a=+0+%2B-+12%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First case,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=a+%2B+12%2F5 ALIGN=MIDDLE  ALT=\"h=a+%2B+12%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then going back through the equations and substituting for h,<BR>\n" );
document.write( "5. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%2B4b=-34 ALIGN=MIDDLE  ALT=\"3a%2B4b=-34\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4h%2B3k%2B7=0 ALIGN=MIDDLE  ALT=\"4h%2B3k%2B7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28a%2B12%2F5%29%2B3k=-7 ALIGN=MIDDLE  ALT=\"4%28a%2B12%2F5%29%2B3k=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B3k=-7-12%2F5 ALIGN=MIDDLE  ALT=\"4a%2B3k=-7-12%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "6.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B3k=-83%2F5 ALIGN=MIDDLE  ALT=\"4a%2B3k=-83%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k-b=%284%2F3%29%2812%2F5%29 ALIGN=MIDDLE  ALT=\"k-b=%284%2F3%29%2812%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "7.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k-b=16%2F5 ALIGN=MIDDLE  ALT=\"k-b=16%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You have three equations, three unknowns, using Cramer's rule,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-2%2F5 ALIGN=MIDDLE  ALT=\"a=-2%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-41%2F5 ALIGN=MIDDLE  ALT=\"h=-41%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-5 ALIGN=MIDDLE  ALT=\"k=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-2%2F5+%2B+12%2F5 ALIGN=MIDDLE  ALT=\"h=-2%2F5+%2B+12%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=10%2F5 ALIGN=MIDDLE  ALT=\"h=10%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the first circle is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%28x-2%29%2B%28y%2B5%29%5E2=16%29 ALIGN=MIDDLE  ALT=\"highlight%28%28x-2%29%2B%28y%2B5%29%5E2=16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "Second case,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=a+-+12%2F5 ALIGN=MIDDLE  ALT=\"h=a+-+12%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again substituting as needed for h,<BR>\n" );
document.write( "8. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3a%2B4b=-34 ALIGN=MIDDLE  ALT=\"3a%2B4b=-34\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28a-12%2F5%29%2B3k%2B7=0 ALIGN=MIDDLE  ALT=\"4%28a-12%2F5%29%2B3k%2B7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B3k=-7%2B48%2F5 ALIGN=MIDDLE  ALT=\"4a%2B3k=-7%2B48%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "9.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B3k=13%2F5 ALIGN=MIDDLE  ALT=\"4a%2B3k=13%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k-b=%284%2F3%29%28-12%2F5%29 ALIGN=MIDDLE  ALT=\"k-b=%284%2F3%29%28-12%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "10.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k-b=-16%2F5 ALIGN=MIDDLE  ALT=\"k-b=-16%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again you have three equations, three unknowns, using Cramer's rule,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=754%2F35 ALIGN=MIDDLE  ALT=\"a=754%2F35\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-863%2F35 ALIGN=MIDDLE  ALT=\"h=-863%2F35\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-195%2F7 ALIGN=MIDDLE  ALT=\"k=-195%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=a+-+12%2F5 ALIGN=MIDDLE  ALT=\"h=a+-+12%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=754%2F35-12%2F5 ALIGN=MIDDLE  ALT=\"h=754%2F35-12%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=134%2F7 ALIGN=MIDDLE  ALT=\"h=134%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the second circle is,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%28x-134%2F7%29%2B%28y%2B195%2F7%29%5E2=16%29 ALIGN=MIDDLE  ALT=\"highlight%28%28x-134%2F7%29%2B%28y%2B195%2F7%29%5E2=16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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