document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1109198>Question 1109198</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of circle passing through (1;-1) and touching he lines 4x+3y+5=0 and 3x-4y-10=0.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #724221 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=724221\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Interesting problem, and quite challenging....<br><BR>\n" );
document.write( "(Although there might be methods of solution that are much easier...!)<br><BR>\n" );
document.write( "If the circle is tangent to the lines 4x+3y+5=0 and 3x-4y-10=0, then the distance from the center of the circle to each of those lines is the same.<BR>\n" );
document.write( "Let the center of the circle be (x,y).<br><BR>\n" );
document.write( "The distance from (x,y) to the line 4x+3y+5=0 is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%284x%2B3y%2B5%29%2Fsqrt%284%5E2%2B3%5E2%29%29+=+abs%28%284x%2B3y%2B5%29%2F5%29 ALIGN=MIDDLE  ALT=\"abs%28%284x%2B3y%2B5%29%2Fsqrt%284%5E2%2B3%5E2%29%29+=+abs%28%284x%2B3y%2B5%29%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The distance from (x,y) to the line 3x-4y-10=0 is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%283x-4y-10%29%2Fsqrt%283%5E2%2B4%5E2%29%29+=+abs%28%283x-4y-10%29%2F5%29 ALIGN=MIDDLE  ALT=\"abs%28%283x-4y-10%29%2Fsqrt%283%5E2%2B4%5E2%29%29+=+abs%28%283x-4y-10%29%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Those distances are the same.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%284x%2B3y%2B5%29%2F5%29+=+abs%28%283x-4y-10%29%2F5%29 ALIGN=MIDDLE  ALT=\"abs%28%284x%2B3y%2B5%29%2F5%29+=+abs%28%283x-4y-10%29%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B3y%2B5+=+3x-4y-10 ALIGN=MIDDLE  ALT=\"4x%2B3y%2B5+=+3x-4y-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B3y%2B5+=+-3x%2B4y%2B10 ALIGN=MIDDLE  ALT=\"4x%2B3y%2B5+=+-3x%2B4y%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B7y%2B15+=+0 ALIGN=MIDDLE  ALT=\"x%2B7y%2B15+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x-y-5+=+0 ALIGN=MIDDLE  ALT=\"7x-y-5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The center of the circle must lie either on line x+7y+15=0 or on line 7x-y-5=0.<br><BR>\n" );
document.write( "A graph of the two given lines and of the two lines on which the center of the circle must lie is appropriate at this point....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-4%2C4%2C-4%2C4%2C%28-4x-5%29%2F3%2C%283x-10%29%2F4%2C%28-x-15%29%2F7%2C7x-5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "red: 4x+3y+5=0<BR>\n" );
document.write( "green: 3x-4y-10=0<BR>\n" );
document.write( "blue: x+7y+15=0<BR>\n" );
document.write( "purple: 7x-y-5=0<br><BR>\n" );
document.write( "Since the circle must contain the point (1,-1), the graph shows us that the center of the circle must be on the purple line (7x-y-5=0) and not the blue line 9x+7y+15=0).<br><BR>\n" );
document.write( "The graph also shows us that there will be two solutions to the problem -- one with the point (1,-1) near the top of a small circle, and another with the point (1,-1) near the bottom of a larger circle.<br><BR>\n" );
document.write( "Let (x,y) = (x,7x-5) be an arbitrary point on line 7x-y-5=0.<br><BR>\n" );
document.write( "We can first verify that any point on that line is equidistant from the two given lines:<br><BR>\n" );
document.write( "The distance from (x,7x-5) to the line 4x+3y+5=0 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%284x%2B3%287x-5%29%2B5%29%2F5%29+=+abs%28%2825x-10%29%2F5%29+=+abs%285x-2%29 ALIGN=MIDDLE  ALT=\"abs%28%284x%2B3%287x-5%29%2B5%29%2F5%29+=+abs%28%2825x-10%29%2F5%29+=+abs%285x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance from (x,7x-5) to the line 3x-4y-10=0 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%283x-4%287x-5%29-10%29%2F5%29+=+abs%28%28-25x%2B10%29%2F5%29+=+abs%28-5x%2B2%29 ALIGN=MIDDLE  ALT=\"abs%28%283x-4%287x-5%29-10%29%2F5%29+=+abs%28%28-25x%2B10%29%2F5%29+=+abs%28-5x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So any point on the line 7x-y-5=0 is equidistant from the two given lines.<BR>\n" );
document.write( "Now we need to find the point(s) on the line where the distance from the point to the given point (1,-1) is the same as the distance to each of the given lines.<br><BR>\n" );
document.write( "Since the distance formula squares the distances, we don't need to concern ourselves with the absolute values.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2+%2B+%28%287x-5%29-%28-1%29%29%5E2+=+%285x-2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2+%2B+%28%287x-5%29-%28-1%29%29%5E2+=+%285x-2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2B1+%2B+49x%5E2-56x%2B16+=+25x%5E2-20x%2B4 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B1+%2B+49x%5E2-56x%2B16+=+25x%5E2-20x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E2-38x%2B13+=+0 ALIGN=MIDDLE  ALT=\"25x%5E2-38x%2B13+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825x-13%29%28x-1%29+=+0 ALIGN=MIDDLE  ALT=\"%2825x-13%29%28x-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+13%2F25 ALIGN=MIDDLE  ALT=\"x+=+13%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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;\" ><br><BR>\n" );
document.write( "For x = 13/25, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+7x-5+=+91%2F25-5+=+-34%2F25 ALIGN=MIDDLE  ALT=\"y+=+7x-5+=+91%2F25-5+=+-34%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" >; the center of that circle is A(13/25,-34/25).<br><BR>\n" );
document.write( "For x=1, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+7x-5+=+2 ALIGN=MIDDLE  ALT=\"y+=+7x-5+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >; the center of that circle is B(1,2).<br><BR>\n" );
document.write( "We can verify that each of the points A(13/25,-34/25) and B(1,2) is equidistant from both of the given lines and from the given point (1,-1).<br><BR>\n" );
document.write( "A(13/25,-34/25)....<BR>\n" );
document.write( "The distance from line 4x+3y+5=0 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%284%2813%2F25%29%2B3%28-34%2F25%29%2B5%29%2F5%29+=+abs%28%2852%2F25-102%2F25%2B5%29%2F5%29+=+abs%28%28-50%2F25%2B5%29%2F5%29+=+abs%28%28-2%2B5%29%2F5%29+=+3%2F5 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance from line 3x-4y-10=0 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%283%2813%2F25%29-4%28-34%2F25%29-10%29%2F5%29+=+abs%28%2839%2F25%2B136%2F25-10%29%2F5%29+=+abs%28%28175%2F25-10%29%2F5%29+=+abs%28%287-10%295%29+=+3%2F5 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance from (1,-1) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%2813%2F25-1%29%5E2%2B%28-34%2F25%2B1%29%5E2%29+=+sqrt%28%2812%2F25%29%5E2%2B%28-9%2F25%29%5E2%29+=+15%2F25+=+3%2F5 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the point (13/25,-34/25) is the center of one circle that satisfies the conditions of the problem.<br><BR>\n" );
document.write( "The equation of that circle is<br><BR>\n" );
document.write( "ANSWER #1: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-13%2F25%29%5E2%2B%28y%2B34%2F25%29%5E2+=+%283%2F5%29%5E2+=+9%2F25 ALIGN=MIDDLE  ALT=\"%28x-13%2F25%29%5E2%2B%28y%2B34%2F25%29%5E2+=+%283%2F5%29%5E2+=+9%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "B(1,2)....<BR>\n" );
document.write( "The distance from line 4x+3y+5=0 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%284%281%29%2B3%282%29%2B5%29%2F5%29+=+abs%28%284%2B6%2B5%29%2F5%29+=+abs%2815%2F5%29+=+3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance from line 3x-4y-10=0 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%283%281%29-4%282%29-10%29%2F5%29+=+abs%28%283-8-10%29%2F5%29+=+abs%28-15%2F5%29+=+3 ALIGN=MIDDLE  ALT=\"abs%28%283%281%29-4%282%29-10%29%2F5%29+=+abs%28%283-8-10%29%2F5%29+=+abs%28-15%2F5%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance from (1,-1) is easily seen to be 3.<br><BR>\n" );
document.write( "So the point (1,2) is the center of a second circle that satisfies the conditions of the problem.<br><BR>\n" );
document.write( "The equation of that circle is<br><BR>\n" );
document.write( "ANSWER #2: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2B%28y-2%29%5E2+=+3%5E2+=+9 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2B%28y-2%29%5E2+=+3%5E2+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Let's add those two circles to our graph to see that they satisfy the conditions of the problem.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-4%2C4%2C-4%2C4%2C%28-4x-5%29%2F3%2C%283x-10%29%2F4%2C7x-5%2Csqrt%28%289%2F25-%28x-13%2F25%29%5E2%29%29-34%2F25%2C-sqrt%28%289%2F25-%28x-13%2F25%29%5E2%29%29-34%2F25%2Csqrt%289-%28x-1%29%5E2%29%2B2%2C-sqrt%289-%28x-1%29%5E2%29%2B2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );