document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200535>Question 1200535</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The line x - 2y + 4 = 0 is tangent to a circle at (0,2). The line y = 2x - 7 is tangent to the same circle at (3, -1). Find the center of the circle.\r<BR>\n" );
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document.write( "NOTE: I WORKED THIS OUT WRONGLY ON PAPER. I DON'T KNOW HOW TO UPLOAD PHOTOS ON THIS MATH SITE WHICH IS VERY LIMITED.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834752 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=834752\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "To share images, upload them to a website like Imgur (or similar). <BR>\n" );
document.write( "After it's uploaded to that website, you can share the link.\r<BR>\n" );
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document.write( "Let's solve for y in the first equation<BR>\n" );
document.write( "x - 2y + 4 = 0<BR>\n" );
document.write( "-2y = -x-4<BR>\n" );
document.write( "y = (-x-4)/(-2)<BR>\n" );
document.write( "y = (1/2)x + 2<BR>\n" );
document.write( "This equation has a slope of 1/2<BR>\n" );
document.write( "The negative reciprocal of 1/2 is -2/1 aka -2<BR>\n" );
document.write( "Therefore, the equation perpendicular to x - 2y + 4 = 0 will have a slope of -2.<BR>\n" );
document.write( "Why are we looking for a perpendicular line? Because the tangent line is perpendicular to the radius when meeting at the point of tangency. Check out the diagram below. \r<BR>\n" );
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document.write( "We'll use this perpendicular slope and the coordinates (x,y) = (0,2) to determine the equation of the perpendicular line is y = -2x+2\r<BR>\n" );
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document.write( "Another approach you can take is outlined in this lesson<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/Linear-equations/perpendicular-line-example1.lesson\">https://www.algebra.com/algebra/homework/Linear-equations/perpendicular-line-example1.lesson</a> \r<BR>\n" );
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document.write( "Through similar steps, the perpendicular line to y = 2x-7 that passes through (3,-1) is y = (-1/2)x + 1/2\r<BR>\n" );
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document.write( "The two perpendicular lines we found<BR>\n" );
document.write( "y = -2x+2<BR>\n" );
document.write( "y = (-1/2)x + 1/2<BR>\n" );
document.write( "will have the circle diameters located on them.\r<BR>\n" );
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document.write( "As such, those lines intersect to pinpoint the center of the circle.\r<BR>\n" );
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document.write( "y = (-1/2)x + 1/2<BR>\n" );
document.write( "-2x+2 = (-1/2)x + 1/2<BR>\n" );
document.write( "2*(-2x+2) = 2*((-1/2)x + 1/2)<BR>\n" );
document.write( "-4x+4 = -x+1<BR>\n" );
document.write( "-4x+x = 1-4<BR>\n" );
document.write( "-3x = -3<BR>\n" );
document.write( "x = -3/(-3)<BR>\n" );
document.write( "x = 1<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "y = -2x+2<BR>\n" );
document.write( "y = -2*1+2<BR>\n" );
document.write( "y = 0<BR>\n" );
document.write( "The center is located at (x,y) = (1,0)<BR>\n" );
document.write( "So we know that (h,k) = (1,0)<BR>\n" );
document.write( "i.e. we have h = 1 and k = 0\r<BR>\n" );
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document.write( "Those h and k values are useful for the circle template<BR>\n" );
document.write( "(x-h)^2 + (y-k)^2 = r^2\r<BR>\n" );
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document.write( "Diagram<BR>\n" );
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document.write( "I used GeoGebra, but Desmos is another good option.\r<BR>\n" );
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document.write( "Answer: Center is <font color=red>(1,0)</font><BR>\n" );
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