Question 1171137
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:**

$$\frac{x^2}{14400} - \frac{y^2}{2500} = 1$$