document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1050916>Question 1050916</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the circle of radius &#8730;29 tangent to the line 2x-5y+16=0 and passing through (9,-3)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #666614 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=666614\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the circle of radius sqrt(29) tangent to the line 2x-5y+16=0 and passing through (9,-3)<BR>\n" );
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document.write( "There are 2 circles that fit.<BR>\n" );
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document.write( "Draw a circle of radius sqrt(29) with its center at (9,-3), call it circle Q.<BR>\n" );
document.write( "The centers of the 2 circles with be on circle Q.<BR>\n" );
document.write( "The centers' distance from the line will be sqrt(29).<BR>\n" );
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document.write( "For the distance from a line:<BR>\n" );
document.write( "d = |A*p + B*q + C|/sqrt(A^2 + B^2) where the line is Ax + By + C = 0 and (p,q) is the point.<BR>\n" );
document.write( "d = |2*p - 5*q + 16|/sqrt(p^2 + q^2) = sqrt(29)<BR>\n" );
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document.write( "That's too complicated.<BR>\n" );
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document.write( "Find a line parallel to the given line and sqrt(29) from it.<BR>\n" );
document.write( "Slope of the given line is 2/5 and the y-int is 16/5<BR>\n" );
document.write( "The slope 2/5 is the tangent of the angle between the line and the x-axis.<BR>\n" );
document.write( "Call the difference in the y-ints z.<BR>\n" );
document.write( "cos(atan(2/5)) = sqrt(29)/z<BR>\n" );
document.write( "z = 5.8<BR>\n" );
document.write( "y-int of the parallel line = 16/5 - 5.8 = -13/5<BR>\n" );
document.write( "Parallel line is y = (2/5)x - 13/5<BR>\n" );
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document.write( "Find the 2 intersections of the line and circle Q.<BR>\n" );
document.write( "Circle Q: (x-9)^2 + (y+3)^2 = 29<BR>\n" );
document.write( "Sub for y<BR>\n" );
document.write( "(x-9)^2 + ((2/5)x - 13/5 +3)^2 = 29<BR>\n" );
document.write( "(x-9)^2 + ((2/5)x + 2/5)^2 = 29<BR>\n" );
document.write( "Multiply by 25 to eliminate fractions.<BR>\n" );
document.write( "(5x-45)^2 + (2x + 2)^2 = 725<BR>\n" );
document.write( "25x^2 - 450x + 2025 + 4x^2 + 8x + 4 = 725<BR>\n" );
document.write( "29x^2 - 442x + 1304 = 0<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29x%5E2%2B-442x%2B1304+=+0 ALIGN=MIDDLE  ALT=\"29x%5E2%2B-442x%2B1304+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-442%29%5E2-4%2A29%2A1304=44100 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-442%29%5E2-4%2A29%2A1304=44100\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=44100 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--442%2B-sqrt%28+44100+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--442%2B-sqrt%28+44100+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-442%29%2Bsqrt%28+44100+%29%29%2F2%5C29+=+11.2413793103448 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-442%29%2Bsqrt%28+44100+%29%29%2F2%5C29+=+11.2413793103448\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-442%29-sqrt%28+44100+%29%29%2F2%5C29+=+4 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-442%29-sqrt%28+44100+%29%29%2F2%5C29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29x%5E2%2B-442x%2B1304 ALIGN=MIDDLE  ALT=\"29x%5E2%2B-442x%2B1304\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=29x%5E2%2B-442x%2B1304+=+%28x-11.2413793103448%29%2A%28x-4%29 ALIGN=MIDDLE  ALT=\"29x%5E2%2B-442x%2B1304+=+%28x-11.2413793103448%29%2A%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 11.2413793103448, 4.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+29%2Ax%5E2%2B-442%2Ax%2B1304+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+29%2Ax%5E2%2B-442%2Ax%2B1304+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "x = 4; y = -1  --> (x-4)^2 + (y+1)^2 = 29 is one circle.<BR>\n" );
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document.write( "x = 326/29; y = 55/29 --> (x - 326/29)^2 + (y - 55/29)^2 = 29<BR>\n" );
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document.write( "email any question.\r<BR>\n" );
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