document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170527>Question 1170527</A>: <I><SPAN STYLE=\"word-break: break-all;\"> LORAN navigational transmitters A and B are located at (-130,0) and (130,0) respectively. A receiver P on a fishing boat somewhere in the first quadrant listens to the pair (A,B) of the transmissions and computes the difference of the distance from boat A and B as 240 miles. Find the equation of the hyperbola on P is located. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851157 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=851157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let A = (-130, 0) and B = (130, 0). Let P = (x, y) be the position of the fishing boat.<BR>\n" );
document.write( "The distance from P to A is $d_A = \sqrt{(x - (-130))^2 + (y - 0)^2} = \sqrt{(x + 130)^2 + y^2}$.<BR>\n" );
document.write( "The distance from P to B is $d_B = \sqrt{(x - 130)^2 + (y - 0)^2} = \sqrt{(x - 130)^2 + y^2}$.\r<BR>\n" );
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document.write( "We are given that the difference in distances is 240 miles, i.e., $|d_A - d_B| = 240$. Since P is in the first quadrant, we can assume $d_A > d_B$, so $d_A - d_B = 240$.\r<BR>\n" );
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document.write( "Thus, $\sqrt{(x + 130)^2 + y^2} - \sqrt{(x - 130)^2 + y^2} = 240$.\r<BR>\n" );
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document.write( "We need to find the equation of the hyperbola.\r<BR>\n" );
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document.write( "First, rearrange the equation:<BR>\n" );
document.write( "$\sqrt{(x + 130)^2 + y^2} = \sqrt{(x - 130)^2 + y^2} + 240$\r<BR>\n" );
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document.write( "Square both sides:<BR>\n" );
document.write( "$(x + 130)^2 + y^2 = (x - 130)^2 + y^2 + 480\sqrt{(x - 130)^2 + y^2} + 240^2$<BR>\n" );
document.write( "$x^2 + 260x + 130^2 + y^2 = x^2 - 260x + 130^2 + y^2 + 480\sqrt{(x - 130)^2 + y^2} + 57600$<BR>\n" );
document.write( "$520x - 57600 = 480\sqrt{(x - 130)^2 + y^2}$\r<BR>\n" );
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document.write( "Divide by 80:<BR>\n" );
document.write( "$6.5x - 720 = 6\sqrt{(x - 130)^2 + y^2}$\r<BR>\n" );
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document.write( "Square both sides again:<BR>\n" );
document.write( "$(6.5x - 720)^2 = 36[(x - 130)^2 + y^2]$<BR>\n" );
document.write( "$42.25x^2 - 9360x + 518400 = 36(x^2 - 260x + 16900 + y^2)$<BR>\n" );
document.write( "$42.25x^2 - 9360x + 518400 = 36x^2 - 9360x + 608400 + 36y^2$<BR>\n" );
document.write( "$6.25x^2 - 36y^2 = 90000$\r<BR>\n" );
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document.write( "Divide by 90000:<BR>\n" );
document.write( "$\frac{6.25x^2}{90000} - \frac{36y^2}{90000} = 1$<BR>\n" );
document.write( "$\frac{x^2}{14400} - \frac{y^2}{2500} = 1$<BR>\n" );
document.write( "$\frac{x^2}{120^2} - \frac{y^2}{50^2} = 1$\r<BR>\n" );
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document.write( "Therefore, the equation of the hyperbola is $\frac{x^2}{14400} - \frac{y^2}{2500} = 1$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\frac{x^2}{14400}-\frac{y^2}{2500}=1}$<BR>\n" );
document.write( " </SPAN>\n" );
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