SOLUTION: LORAN navigational transmitters A and B are located at (-130,0) and (130,0) respectively. A receiver P on a fishing boat somewhere in the first quadrant listens to the pair (A,B) o

Question 1170527: LORAN navigational transmitters A and B are located at (-130,0) and (130,0) respectively. A receiver P on a fishing boat somewhere in the first quadrant listens to the pair (A,B) of the transmissions and computes the difference of the distance from boat A and B as 240 miles. Find the equation of the hyperbola on P is located.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let A = (-130, 0) and B = (130, 0). Let P = (x, y) be the position of the fishing boat.
The distance from P to A is $d_A = \sqrt{(x - (-130))^2 + (y - 0)^2} = \sqrt{(x + 130)^2 + y^2}$.
The distance from P to B is $d_B = \sqrt{(x - 130)^2 + (y - 0)^2} = \sqrt{(x - 130)^2 + y^2}$.
We are given that the difference in distances is 240 miles, i.e., $|d_A - d_B| = 240$. Since P is in the first quadrant, we can assume $d_A > d_B$, so $d_A - d_B = 240$.
Thus, $\sqrt{(x + 130)^2 + y^2} - \sqrt{(x - 130)^2 + y^2} = 240$.
We need to find the equation of the hyperbola.
First, rearrange the equation:
$\sqrt{(x + 130)^2 + y^2} = \sqrt{(x - 130)^2 + y^2} + 240$
Square both sides:
$(x + 130)^2 + y^2 = (x - 130)^2 + y^2 + 480\sqrt{(x - 130)^2 + y^2} + 240^2$
$x^2 + 260x + 130^2 + y^2 = x^2 - 260x + 130^2 + y^2 + 480\sqrt{(x - 130)^2 + y^2} + 57600$
$520x - 57600 = 480\sqrt{(x - 130)^2 + y^2}$
Divide by 80:
$6.5x - 720 = 6\sqrt{(x - 130)^2 + y^2}$
Square both sides again:
$(6.5x - 720)^2 = 36[(x - 130)^2 + y^2]$
$42.25x^2 - 9360x + 518400 = 36(x^2 - 260x + 16900 + y^2)$
$42.25x^2 - 9360x + 518400 = 36x^2 - 9360x + 608400 + 36y^2$
$6.25x^2 - 36y^2 = 90000$
Divide by 90000:
$\frac{6.25x^2}{90000} - \frac{36y^2}{90000} = 1$
$\frac{x^2}{14400} - \frac{y^2}{2500} = 1$
$\frac{x^2}{120^2} - \frac{y^2}{50^2} = 1$
Therefore, the equation of the hyperbola is $\frac{x^2}{14400} - \frac{y^2}{2500} = 1$.
Final Answer: The final answer is $\boxed{\frac{x^2}{14400}-\frac{y^2}{2500}=1}$

RELATED QUESTIONS
Navigational transmitters Q and R are located at (-130,0) and (130,0) respectively. A... (answered by CPhill)

Two radio transmitters are positioned along the coast, 500 miles apart. A signal is sent... (answered by richwmiller)

How do you find the x and y intercept of y-2x=5, a. x-intercept located at (� , 0);... (answered by checkley71)

I've got two LORAN stations A and B that are 500 miles apart. A and B are also the Foci... (answered by rothauserc)

d. According to an officer in the Health Care Financing Administration, the average... (answered by textot)

A satellite dish is shaped like a paraboloid of revolution. This means that it can be... (answered by onyulee)

IQs are normally distributed with mean 100 and standard deviation 16. a) Less than 110 (answered by stanbon)

A(1,2), B(-1,1), C(1,0), D(-1,0) and P is (t,0). (cordinates of p are located between 0... (answered by solver91311)

It is given that the formula is C=4h+9f+4p: C=calories h=carbohydrates f=fat p=protein (answered by Fombitz)