Trig Help???? Hyperbolas?

Question

Two LORAN stations are placed 150 miles apart along a straight shore and act as the foci of a hyperbola. A ship records a time difference of 0.0005 second between the LORAN signals. Set up a rectangular coordinate system to determine where the ship would reach shore if it were to follow the hyperbola corresponding to this time difference. The speed of radio-signals is 186000 miles per second.










What is the equation of the hyperbola?



##6 (20 points). For the hyperbola 4x2 - 25y2=100, answer the following:


(a) What are the coordinates of the left vertex?



What are the coordinates of the right focus?




Write the equations for both asymptotes.

PLEASE SHOW ME HOW YOU DID IT NOT JUST ANSWERS

Answer 1

First of the 2 problems:

Let (0, 0) is the center of hyperbola.
Foci of the hyperbola are (-75, 0) and (75, 0).

The distance between vertexes of hyperbola is
186000 m/s * 0.0005 s = 93 miles.

Vertexes are (-46.5, 0) and (46.5, 0).
The ship would reach shore in one of these 2 vertexes.

Let a, b and c are parameters of hyperbola according to
http://mathworld.wolfram.com/Hyperbola.html

a = 46.5
c = 75
b² = c² - a²
b² = 3462,75

The equation of the hyperbola is
x²/a² - y²/b² = 1
x²/2162,25 - y²/3462,75 = 1

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Answer 2

A conventional form of a hyperbola is (x^2/a^2) - (y^2/b^2) = a million and you ought to discover a and b. we are able to discover a from the factor (2, 0). 4/a^2 = a million a^2 = 4 we are able to discover b from understanding that for extremely great x and y, y = 2x (x^2/4) - (4x^2/b^2) = a million Multiply by using 4b^2: x^2b^2 - 16x^2 = 4b^2 x^2(b^2 - sixteen) = 4b^2 {(b^2 - sixteen) / 4b^2} = a million / x^2 for the reason that x is extremely great, {(b^2 - sixteen) / 4b^2} = 0 b^2 - sixteen = 0 b^2 = sixteen the main suitable equation is (x^2 / 4) - (y^2 / sixteen) = a million