document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209537>Question 1209537</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let k be a positive real number. The line x + y = k/2 and the circle x^2 + y^2 = 3x - 6y + k are drawn. Find k so that the line is tangent to the circle.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849317 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849317\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the value of k for which the line x + y = k/2 is tangent to the circle x² + y² = 3x - 6y + k:\r<BR>\n" );
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document.write( "**1. Rewrite the circle equation:**\r<BR>\n" );
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document.write( "Complete the square for both x and y terms in the circle equation:\r<BR>\n" );
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document.write( "x² - 3x + y² + 6y = k<BR>\n" );
document.write( "(x² - 3x + 9/4) + (y² + 6y + 9) = k + 9/4 + 9<BR>\n" );
document.write( "(x - 3/2)² + (y + 3)² = k + 45/4\r<BR>\n" );
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document.write( "This shows that the circle has center (3/2, -3) and radius √(k + 45/4).\r<BR>\n" );
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document.write( "**2. Express the line in terms of y:**\r<BR>\n" );
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document.write( "The line equation x + y = k/2 can be rewritten as y = -x + k/2.\r<BR>\n" );
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document.write( "**3. Condition for tangency:**\r<BR>\n" );
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document.write( "A line is tangent to a circle if the distance from the center of the circle to the line is equal to the radius of the circle.\r<BR>\n" );
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document.write( "**4. Distance from a point to a line:**\r<BR>\n" );
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document.write( "The distance from a point (x₁, y₁) to a line ax + by + c = 0 is given by:\r<BR>\n" );
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document.write( "Distance = |ax₁ + by₁ + c| / √(a² + b²)\r<BR>\n" );
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document.write( "In our case, the point is (3/2, -3) and the line is x + y - k/2 = 0.  So, a=1, b=1, c=-k/2, x₁=3/2, and y₁=-3.\r<BR>\n" );
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document.write( "Distance = |(1)(3/2) + (1)(-3) - k/2| / √(1² + 1²)<BR>\n" );
document.write( "Distance = |3/2 - 3 - k/2| / √2<BR>\n" );
document.write( "Distance = |-3/2 - k/2| / √2<BR>\n" );
document.write( "Distance = |(-3 - k)/2| / √2<BR>\n" );
document.write( "Distance = |k + 3| / (2√2)\r<BR>\n" );
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document.write( "**5. Set distance equal to radius:**\r<BR>\n" );
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document.write( "Now, set the distance equal to the radius:\r<BR>\n" );
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document.write( "|k + 3| / (2√2) = √(k + 45/4)\r<BR>\n" );
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document.write( "**6. Solve for k:**\r<BR>\n" );
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document.write( "Square both sides to get rid of the square roots:\r<BR>\n" );
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document.write( "(k + 3)² / 8 = k + 45/4<BR>\n" );
document.write( "(k² + 6k + 9) / 8 = (4k + 45) / 4<BR>\n" );
document.write( "k² + 6k + 9 = 2(4k + 45)<BR>\n" );
document.write( "k² + 6k + 9 = 8k + 90<BR>\n" );
document.write( "k² - 2k - 81 = 0\r<BR>\n" );
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document.write( "Use the quadratic formula to solve for k:\r<BR>\n" );
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document.write( "k = (-b ± √(b² - 4ac)) / 2a<BR>\n" );
document.write( "k = (2 ± √((-2)² - 4(1)(-81))) / 2(1)<BR>\n" );
document.write( "k = (2 ± √(4 + 324)) / 2<BR>\n" );
document.write( "k = (2 ± √328) / 2<BR>\n" );
document.write( "k = (2 ± 2√82) / 2<BR>\n" );
document.write( "k = 1 ± √82\r<BR>\n" );
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document.write( "Since k must be positive, we take the positive solution:\r<BR>\n" );
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document.write( "k = 1 + √82\r<BR>\n" );
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document.write( "Therefore, the value of k for which the line is tangent to the circle is 1 + √82.<BR>\n" );
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