document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171137>Question 1171137</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Navigational transmitters Q and R are located at (-130,0) and (130,0) respectively. A receiver A on a fishing boat somewhere in the first quadrant listens to pair (Q,R) of the transmissions and computes the difference of the distance from boat Q and R as 240 miles. What is the equation of the hyperbola on which A 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 #850989 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=850989\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**Understanding the Hyperbola**\r<BR>\n" );
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document.write( "The key information is that the difference of the distances from the receiver A to the transmitters Q and R is constant (240 miles). This defines a hyperbola.\r<BR>\n" );
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document.write( "**Given Information:**\r<BR>\n" );
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document.write( "* Transmitter Q: (-130, 0)<BR>\n" );
document.write( "* Transmitter R: (130, 0)<BR>\n" );
document.write( "* Difference in distances: |AQ - AR| = 240 miles<BR>\n" );
document.write( "* Receiver A is in the first quadrant.\r<BR>\n" );
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document.write( "**Hyperbola Equation**\r<BR>\n" );
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document.write( "The standard equation of a hyperbola with horizontal transverse axis and center at the origin is:\r<BR>\n" );
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document.write( "$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* 2a is the difference of the distances from any point on the hyperbola to the foci.<BR>\n" );
document.write( "* The foci are at (±c, 0).<BR>\n" );
document.write( "* c is the distance from the center to each focus.<BR>\n" );
document.write( "* b^2 = c^2 - a^2\r<BR>\n" );
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document.write( "**Finding the Parameters**\r<BR>\n" );
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document.write( "1.  **2a:** The difference in distances is 240 miles, so 2a = 240, which means a = 120.\r<BR>\n" );
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document.write( "2.  **c:** The foci are at (-130, 0) and (130, 0), so c = 130.\r<BR>\n" );
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document.write( "3.  **b^2:** We can find b^2 using the relationship b^2 = c^2 - a^2:<BR>\n" );
document.write( "    * b^2 = 130^2 - 120^2 = 16900 - 14400 = 2500\r<BR>\n" );
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document.write( "**Writing the Equation**\r<BR>\n" );
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document.write( "Now, we can substitute the values of a^2 and b^2 into the hyperbola equation:\r<BR>\n" );
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document.write( "$$\frac{x^2}{120^2} - \frac{y^2}{2500} = 1$$\r<BR>\n" );
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document.write( "$$\frac{x^2}{14400} - \frac{y^2}{2500} = 1$$\r<BR>\n" );
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document.write( "**Therefore, the equation of the hyperbola on which A is located is:**\r<BR>\n" );
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document.write( "$$\frac{x^2}{14400} - \frac{y^2}{2500} = 1$$<BR>\n" );
document.write( " </SPAN>\n" );
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