1. \[ \frac{x^2}{25} – \frac{y^2}{24} = 1 \]
2. \[ \frac{x^2}{25} – \frac{y^2}{144} = 1 \]
3. \[ \frac{y^2}{4} – \frac{x^2}{32} = 1 \]
4. \[ \frac{y^2}{16} – \frac{x^2}{209} = 1 \]
5. \[ \frac{ (x – 3)^2 }{9} – \frac{ (y – 2)^2 }{16} = 1 \]
6. \[ \frac{(y + 3)^2}{9} – \frac{ (x + 3)^2 }{27} = 1 \]
7. \[ \frac{ (x – 2)^2 }{4} – \frac{(y – 2)^2}{5} = 1 \]
8. \[ \frac{x^2}{\frac{9}{5}} – \frac{y^2}{\frac{36}{5}} = 1 \]
9. \[ \frac{y^2}{50} – \frac{x^2}{8} = 1 \]
10. \[ \frac{y^2}{4} – \frac{x^2}{12} = 1 \]
11. \[ \frac{(x – 4)^2}{2} – \frac{(y – 2)^2}{2} = 1 \]
12. \[ \frac{(y – 2)^2}{3} – \frac{ \left( x – \frac{1}{2} \right)^2 }{ \frac{3}{4} } = 1 \]
13. \[ \frac{ (y + 2)^2 }{9} – \frac{(x – 3)^2}{36} = 1 \]
14. \[ \frac{(x – 1)^2}{4} – \frac{(y – 3)^2}{5} = 1 \]
15. \[ \frac{ (y – 1)^2 }{9} – \frac{(x + 2)^2}{3} = 1 \]
16.  
    1. \[ \frac{ (x – 3)^2 }{16} – \frac{(y – 2)^2}{9} = 1 \]
    2. \[ (-1, \, 2), \, (7, \, 2) \]
    3. \[ (-2, \, 2), \, (8, \, 2) \]
    4. \(3x – 4y – 1 = 0\);   \(3x + 4y – 17 = 0\)
17.  
    1. \[ \frac{(y – 2)^2}{4} – \frac{(x + 4)^2}{9} = 1 \]
    2. \[ (-4, \, 0), \, (-4, \, 4) \]
    3. \(\left( -4, \, 2 – \sqrt{13} \right)\),   \(\left( -4, \, 2 + \sqrt{13} \right)\)
    4. \(3y – 2x – 14 = 0\);   \(3y + 2x + 2 = 0\)
18.  
    1. \[ \frac{(x – 1)^2}{4} – \frac{(y + 3)^2}{64} = 1 \]
    2. \[ (-1, \, -3), \, (3, \, -3) \]
    3. \(\left( 1 – 2 \sqrt{17}, \, -3 \right)\),   \(\left( 1 + 2\sqrt{17}, \, -3 \right)\)
    4. \(y – 4x + 7 = 0\);   \(y + 4x – 1 = 0\)
19.  
    1. \[ \frac{(x – 2)^2}{9} – \frac{ (y – 3)^2 }{4} = 1 \]
    2. \[ (-1, \, 3), \, (5, \, 3) \]
    3. \[ \left( 2 – \sqrt{13}, \, 3 \right), \, \left( 2 + \sqrt{13}, \, 3 \right) \]
    4. \(2x – 3y + 5 = 0\);   \(2x + 3y – 13 = 0\)
20.  
    1. \[ \frac{x^2}{3,600} – \frac{y^2}{6,400} = 1 \]
    2. \(40 \, km\) away from the station \(B\).
    3. \(0.0002 \; sec.\)
21. \[ \frac{x^2}{28,900} – \frac{y^2}{11,100} = 1 \]
22. \[ \frac{x^2}{3} – \frac{ (y – 1)^2 }{1} = 1 \]

In exercises 1 through 15, find the standard equation of the hyperbola satisfying the given conditions.

1. Foci: \((\pm 7, 0)\). Vertices: \((\pm 5, 0)\).
2. Foci: \((\pm 13, 0)\). Vertices: \((\pm 5, 0)\).
3. Foci: \((0, \pm6)\). Vertices: \((0, \pm2)\).
4. Foci: \((0, \pm 15)\). Vertices: \((0, \pm4)\).
5. Foci: (-2, 2), (8, 2). Vertices: (0, 2), (6, 2).
6. One focus: (-3, 3). Vertices: (-3, 0), (-3, -6 ).
7. Foci: (-1,2), (5, 2). One vertex: (4, 2).
8. Foci: \((\pm 3, 0)\). Asymptotes: \(y = \pm 2x\).
9. Foci: \((0,\, \pm\sqrt{58})\). Asymptotes: \(y = \pm \cfrac{5}{2}x\).
10. Asymptotes: \(x = \pm \sqrt{3}y\). Passes trough \((6, 4)\).
11. Foci: (2, 2), (6, 2), Asymptotes: \(y = x + 2\), \(y = -x + 6\).
12. Asymptotes: \(y = 2x + 1\), \(y = -2x + 3\). Passes trough (0, 0).
13. Vertices: (-5, 3), (1, 3). One asymptote: \(2y – x + 7 = 0\).
14. Vertices: (-1, 3), (3, 3). Eccentricity: \(e=\cfrac{3}{2}\).
15. Vertices: (-2, -2), (-2, 4). Latus rectum: \(L=2\).

In the exercises 16 through 19, completing squares, find:

1. the standard form of the hyperbola.
2. the vertices.
3. the foci.
4. the asymptotes.

16. \[ 9x^2 – 16y^2 -54x + 64y – 127 = 0 \]
    \[ \begin{aligned} 9x^2 – 16y^2 -54x &+ 64y \\[1em] &- 127 = 0 \end{aligned} \]
17. \[ 4x^2 – 9y^2 + 32x + 36y + 64 = 0 \]
18. \[ 16x^2 – y^2 – 32x – 6y- 57 = 0 \]
19. \[ 4x^2 – 9y^2 – 16x + 54y – 101 = 0 \]
    \[ \begin{aligned} 4x^2 – 9y^2 – 16x &+ 54y \\[1em] &- 101 = 0 \end{aligned} \]

20. Two stations \(A\) and \(B\) of a LORAN system are 200 \(km\) away on a straight coast. Station B is to the west, and station A is to the east.

    One ship receives the signal from the station \(B\), 0.0004 seconds before the signal from the station \(A\). The speed of the signal is 300,000 \(km/sec\).

    

    1. If the ship sails to the coast keeping a constant time difference, find the equation of the route.
    2. At what point will the ship reach the coast?
    3. If the dock is located between the two stations, 70 \(km\) away from the station \(B\), find the time difference the ship must keep.
21. Two observers are stationed at points \(F_1=(-200, 0)\) and \(F_2=(200, 0)\) on the XY plane. An explosion occurs at a certain point on the XY plane and is heard by the observer at \(F_2\) one second before it is heard by the observer at \(F_1\). Determine the equation of the hyperbola where the explosion occurred.
22. Find the equation of the set of points \(P\) in the plane such that the distance from \(P\) to the point (−2, 1) is \(\frac{2}{\sqrt{3}}\) times the distance from \(P\) to the line \(x = -\frac{3}{2}\).

23. Let \(P=(x_1, y_1)\) be a point of the hyperbola \(\cfrac{x^2}{a^2}-\cfrac{y^2}{b^2}=1\).

    1. Prove that the slope of the tangent line to the hyperbola at the point \(P = (x_1, y_1)\) is:

       \[ m=\frac{b^2 x_1}{a^2 y_1} \]

       Hint: The line \(T\): \(y = m(x-x_1) + y_1\) pass through \(P = (x_1, y_1)\). \(T\) is tangent to the hyperbola if they intercept at a single point. Follow the same steps given in the solved problem 3.2.6.

       Another Hint: Wait until you learn derivatives.

    2. Prove that the a equation of the tangent line \(T\) to the hyperbola at the point \(P=(x_1, y_1)\) is:

       \[ \cfrac{x_1 x}{a^2}-\cfrac{y_1 y}{b^2}=1 \]