CLEMSON ALGEBRA PROJECT UNIT 14: CONIC SECTIONS
PROBLEM 1: LORAN - LONG-DISTANCE RADIO NAVIGATION LORAN, long-distance radio navigation for aircraft and ships, uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola which has transmitting stations located at the foci. Assume that two stations, 300 miles apart, are positioned on a rectangular coordinate system at points (-150, 0) and (150, 0) and that a ship is traveling on a path with coordinates (x, 75). A. Find the x-coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 microseconds (0.001 second). B. Write the equation of the hyperbola on which the ship is located.
REFERENCE: Precalculus, Third Edition, by Larson and Hostetler, D.C. Heath and Company, 1993.
MATERIALS Casio CFX-9850Ga Plus or ALGEBRA FX2.0 Graphing Calculator
Copyright 1999 by Clemson U. & Casio, Inc.

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Clemson Algebra Project
CONICS ONE SOLUTION TO PROBLEM 1: LORAN A. Find the x-coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 microseconds (0.001 second). The difference in the distance from a point on the hyperbola to the foci is constant. In this instance, use the time difference in receipt of the pulses to determine that constant difference. Time travels at 186,000 miles per second. Because the time difference is 0.001 second, the constant difference must be.001 seconds times 186,000 miles per second, or 186 miles. Next, well use the distance formula to find an expression that describes the distance from each of the foci to the location of the ship. The equation is: ( x + 150) 2 + ( x 150) 2 + = 186
To solve, this equation, we will use the Equation mode on the calculator. From the MAIN MENU, choose Equation, and then: x x Press F3 for the Solver. Type in the equation, being careful with parentheses make sure you include parentheses around the entire expressions under the radicals. See below left for the beginning of the equation. (NOTE: Your calculator may show something different for X at this point.) x Press F6 to solve the equation. See below right.
Our solution is X = 110.28. This tells us that the ship is located at the point which has coordinates (110.28, 75).

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CONICS B. Write the equation of the hyperbola on which the ship is located. The general form for our hyperbola is x2 y2 = 1. The relationship between a2 b2
a (the distance from the center to a vertex), b (the distance from the center to the conjugate axis), and c (the distance from the center to a focus) is a2 + b2 = c2. Consequently, b 2 = c 2 a 2. We already know that the foci are at (-150, 0) and (150, 0), giving us a value of 150 for c. Substituting, we have that b 2 = a 2. Making another substitution, the equation for our hyperbola becomes y2 x2 = 1. We can now substitute the x and y values of the point we found a 2 a 2 in part A into this equation. This gives us an equation with a as the only variable, 110.= 1. a2 a 2 Once again, we will use the equation solving capabilities of the calculator. From the MAIN MENU, call up Equations. Then, x x Press F3 for Solver. Type in the equation. Be careful with parentheses, and use x instead of a. Press EXE. See below left for part of this. x Press F6 to solve the equation.

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CONICS We now have a = 93.36 and c = 150. From the MAIN MENU, choose Run, and find 93.to find b. You should get b = 117.40. Thus our equation is

y2 x2 =1. 93.117.To graph this hyperbola, from the MAIN MENU, call up Conics. Then, x Set the viewing window by pressing SHIFT F3. Remember to press
EXE after each entry. One possible window is shown below left. Press EXIT when finished. x Then, use the down arrow to highlight the horizontal hyperbola and press EXE. Type in 93. 36 for A, 117.4 for B, 0 for H, and 0 for K, pressing EXE after each entry. x Press F6 to draw the graph. See below right.
You can trace points on the hyperbola by pressing F1 and using the arrow keys.
If you press F5 you will access the Graph Solver. Note you can find the foci, the intercepts, and the vertices with this.
To draw asymptotes, press F5 twice.

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