document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170541>Question 1170541</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two stations, located at M(−1.5,0) and N(1.5,0)(units are in km), simultaneously send sound signals to a ship, with the signal traveling at the speed of 0.33 km/s. If the signal from N was received by the ship four seconds before the signal it received from M, find the equation of the curve containing the possible location of the ship. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851153 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=851153\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the ship's location be (x, y).<BR>\n" );
document.write( "The distance between the ship and station M is $d_M = \sqrt{(x + 1.5)^2 + y^2}$.<BR>\n" );
document.write( "The distance between the ship and station N is $d_N = \sqrt{(x - 1.5)^2 + y^2}$.\r<BR>\n" );
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document.write( "The time it takes for the signal from M to reach the ship is $t_M = \frac{d_M}{0.33}$.<BR>\n" );
document.write( "The time it takes for the signal from N to reach the ship is $t_N = \frac{d_N}{0.33}$.\r<BR>\n" );
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document.write( "We are given that the signal from N was received 4 seconds before the signal from M. This means $t_M - t_N = 4$.<BR>\n" );
document.write( "Substituting the expressions for $t_M$ and $t_N$, we get:<BR>\n" );
document.write( "$$\frac{d_M}{0.33} - \frac{d_N}{0.33} = 4$$<BR>\n" );
document.write( "$$d_M - d_N = 4(0.33) = 1.32$$<BR>\n" );
document.write( "$$\sqrt{(x + 1.5)^2 + y^2} - \sqrt{(x - 1.5)^2 + y^2} = 1.32$$\r<BR>\n" );
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document.write( "Now, we need to eliminate the square roots.<BR>\n" );
document.write( "$$\sqrt{(x + 1.5)^2 + y^2} = \sqrt{(x - 1.5)^2 + y^2} + 1.32$$<BR>\n" );
document.write( "Square both sides:<BR>\n" );
document.write( "$$(x + 1.5)^2 + y^2 = (x - 1.5)^2 + y^2 + 2(1.32)\sqrt{(x - 1.5)^2 + y^2} + (1.32)^2$$<BR>\n" );
document.write( "$$x^2 + 3x + 2.25 + y^2 = x^2 - 3x + 2.25 + y^2 + 2.64\sqrt{(x - 1.5)^2 + y^2} + 1.7424$$<BR>\n" );
document.write( "$$6x - 1.7424 = 2.64\sqrt{(x - 1.5)^2 + y^2}$$<BR>\n" );
document.write( "Divide by 2.64:<BR>\n" );
document.write( "$$\frac{6x - 1.7424}{2.64} = \sqrt{(x - 1.5)^2 + y^2}$$<BR>\n" );
document.write( "$$\frac{600x - 174.24}{264} = \sqrt{(x - 1.5)^2 + y^2}$$<BR>\n" );
document.write( "$$\frac{250x - 72.6}{110} = \sqrt{(x - 1.5)^2 + y^2}$$<BR>\n" );
document.write( "$$\frac{125x - 36.3}{55} = \sqrt{(x - 1.5)^2 + y^2}$$<BR>\n" );
document.write( "Square both sides again:<BR>\n" );
document.write( "$$\left(\frac{125x - 36.3}{55}\right)^2 = (x - 1.5)^2 + y^2$$<BR>\n" );
document.write( "$$\frac{(125x - 36.3)^2}{55^2} = (x - 1.5)^2 + y^2$$<BR>\n" );
document.write( "$$\frac{15625x^2 - 9075x + 1317.69}{3025} = x^2 - 3x + 2.25 + y^2$$<BR>\n" );
document.write( "$$15625x^2 - 9075x + 1317.69 = 3025x^2 - 9075x + 6806.25 + 3025y^2$$<BR>\n" );
document.write( "$$12600x^2 - 3025y^2 = 5488.56$$<BR>\n" );
document.write( "Divide by 5488.56:<BR>\n" );
document.write( "$$\frac{12600x^2}{5488.56} - \frac{3025y^2}{5488.56} = 1$$<BR>\n" );
document.write( "$$\frac{x^2}{0.4356} - \frac{y^2}{1.8144} = 1$$<BR>\n" );
document.write( "$$\frac{x^2}{(0.66)^2} - \frac{y^2}{(1.347)^2} = 1$$\r<BR>\n" );
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document.write( "The equation of the curve is a hyperbola.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\frac{x^2}{0.4356}-\frac{y^2}{1.8144}=1}$<BR>\n" );
document.write( " </SPAN>\n" );
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