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Question 1171137: 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?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**Understanding the Hyperbola**
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.
**Given Information:**
* Transmitter Q: (-130, 0)
* Transmitter R: (130, 0)
* Difference in distances: |AQ - AR| = 240 miles
* Receiver A is in the first quadrant.
**Hyperbola Equation**
The standard equation of a hyperbola with horizontal transverse axis and center at the origin is:
$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
Where:
* 2a is the difference of the distances from any point on the hyperbola to the foci.
* The foci are at (±c, 0).
* c is the distance from the center to each focus.
* b^2 = c^2 - a^2
**Finding the Parameters**
1. **2a:** The difference in distances is 240 miles, so 2a = 240, which means a = 120.
2. **c:** The foci are at (-130, 0) and (130, 0), so c = 130.
3. **b^2:** We can find b^2 using the relationship b^2 = c^2 - a^2:
* b^2 = 130^2 - 120^2 = 16900 - 14400 = 2500
**Writing the Equation**
Now, we can substitute the values of a^2 and b^2 into the hyperbola equation:
$$\frac{x^2}{120^2} - \frac{y^2}{2500} = 1$$
$$\frac{x^2}{14400} - \frac{y^2}{2500} = 1$$
**Therefore, the equation of the hyperbola on which A is located is:**
$$\frac{x^2}{14400} - \frac{y^2}{2500} = 1$$
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