document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=597160>Question 597160</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone help me with this question please?\r<BR>\n" );
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document.write( "A circle C passes through the point (-12, 9) and is given by the equation<BR>\n" );
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document.write( "If the equation of the tangent to the given circle at the point (-4, 1) is given by:<BR>\n" );
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document.write( "Find the values of a,b and r.\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #378046 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=378046\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "This looks a good deal uglier than it actually is.  The fortunate placement of the the point (-12,9) makes things much easier, as you will see.\r<BR>\n" );
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document.write( "Step 1:  Take the given tangent line equation and put it into slope-intercept form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ x\ +\ 5\">\r<BR>\n" );
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document.write( "noting that the slope is 1.\r<BR>\n" );
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document.write( "Step 2: Use the fact that a radius to a point of tangency is perpendicular to the tangent line at that point, perpendicular lines have negative reciprocal slopes, and the point-slope form to write an equation of the line containing the radius from the center of the circle to the point of tangency.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ 1\ =\ -1(x\ +\ 4)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ -x\ -\ 3\">\r<BR>\n" );
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document.write( "Step 3:  Here is the fortunate circumstance.  Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 9\ =\ -(-12)\ -\ 3\">, the point (-12, 9) <i><b>is the other end of the diameter</b></i> contained in the line <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\ =\ -x\ -\ 3\">.  Therefore the center of the circle must be the midpoint of the segment with endpoints (-12, 9) and (-4, 1)!\r<BR>\n" );
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document.write( "Step 4:  Calculate the midpoint coordinates using the midpoint formulas:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_m\ = \frac{x_1 + x_2}{2}\ =\ \frac{-12 + (-4)}{2}\ =\ -8\"><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_m\ = \frac{y_1 + y_2}{2}\ = \frac{9 + 1}{2}\ =\ 5\">\r<BR>\n" );
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document.write( "Hence the midpoint of the diameter which must be the center of the circle is (-8, 5).\r<BR>\n" );
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document.write( "Step 5: The distance from either endpoint to the center is the radius, so using the distance formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ sqrt{(x_1\ -\ x_2)^2\ +\ (y_1\ -\ y_2)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ sqrt{(-4\ -\ (-8))^2\ +\ (1\ -\ 5)^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ sqrt{32}\ =\ 4\sqrt{2}\">\r<BR>\n" );
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document.write( "Step 6:  The equation of a circle with center at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h,k)\"> and radius <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE r\"> is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x\ -\ h)^2\ +\ (y\ -\ k)^2\ =\ r^2\">, so read your values directly:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ -8\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ b\ =\ 5\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ r\ =\ 4\sqrt{2}\">\r<BR>\n" );
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document.write( "And the final form of the equation of your circle is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ +\ 8)^2\ +\ (y\ -\ 5)^2\ =\ 32\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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