document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1024077>Question 1024077</A>: <I><SPAN STYLE=\"word-break: break-all;\"> for what k will the line y=x+k be the tangent to the hyperbola x^2-4y^2=48? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #639570 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=639570\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x = y-k.\r<BR>\n" );
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document.write( "==>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4y%5E2+=+%28y-k%29%5E2-4y%5E2+=+-3y%5E2+-+2ky+%2B+k%5E2+=+48 ALIGN=MIDDLE  ALT=\"x%5E2-4y%5E2+=+%28y-k%29%5E2-4y%5E2+=+-3y%5E2+-+2ky+%2B+k%5E2+=+48\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y%5E2+%2B+2ky+%2B+48+-+k%5E2+=+0 ALIGN=MIDDLE  ALT=\"3y%5E2+%2B+2ky+%2B+48+-+k%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For tangency, the value of 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;\" > must be 0:\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k%29%5E2+-4%283%29%2848-k%5E2%29+=+0 ALIGN=MIDDLE  ALT=\"%282k%29%5E2+-4%283%29%2848-k%5E2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4k%5E2+-+12%2848-k%5E2%29+=+0 ALIGN=MIDDLE  ALT=\"4k%5E2+-+12%2848-k%5E2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4k%5E2+-+144+=+0 ALIGN=MIDDLE  ALT=\"4k%5E2+-+144+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > after simplifying a little further...\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2+=+36 ALIGN=MIDDLE  ALT=\"k%5E2+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or k = 6, -6.\r<BR>\n" );
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document.write( "Thus the lines y = x+6 and y = x - 6 are tangent to the hyperbola.\r<BR>\n" );
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