How to find equation of hyperbola given foci and a point?
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I am currently studying multivariate calculus at university, and ive been given some practice problems before the first assignment.
The problem is:
A hyperbola may be defined as the set of points in a plane, the
difference of whose distances from two fixed points $F_1$ and $F_2$ is a
constant. Let P be a point on the hyperbola. Suppose the foci of the
hyperbola are located at (0, ±c), and that $|P F_1| − |P F_2| = ±2a$. It
may be shown that the equation of the hyperbola is given by
$frac{y^2}{a^2} - frac{x^2}{b^2} = 1, where space c^2 = a^2 + b^2$
Hyperbolas have many useful applications, one of which is their use in
navigation systems to determine the location of a ship. Two
transmitting stations, with known positions transmit radio signals to
the ship. The difference in the reception times o
f the signals is used to
compute the difference in distance between the ship and the two transmitting stations. This
infomation places the ship on a hyperbola whose foci are the transmitting stations.
Suppose that radio stations are located at Tanga and Dar es Salaam, two cities on the
north-south coastline of Tanzania. Dar-es Salaam is located 200 km due south of Tanga (you
may assume that Dar es Salaam is directly south of Tanga). Simultaneous radio signals are
transmitted from Tanga and Dar es Salaam to a ship in the Indian Ocean. The ship receives
the signal from Tanga 500 microseconds (µs) before it receives the signal from Dar es Salaam.
Assume that the speed of radio signals is 300m/µs.
(a) By setting up an xy-coordinate system with Tanga having
coordinates (0, 100), determine the equation of the hyperbola on which
the ship lies.
(b) Given that the ship is due east of
Tanga, determine the coordinates of the ship.
If someone wouldnt mind giving me a few hints as to how I could solve this, I would be very grateful.
Thanks
Tim
conic-sections
asked Aug 15 '15 at 8:40
FishingfonFishingfon
3012718
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add a comment |
$begingroup$
I am currently studying multivariate calculus at university, and ive been given some practice problems before the first assignment.
The problem is:
A hyperbola may be defined as the set of points in a plane, the
difference of whose distances from two fixed points $F_1$ and $F_2$ is a
constant. Let P be a point on the hyperbola. Suppose the foci of the
hyperbola are located at (0, ±c), and that $|P F_1| − |P F_2| = ±2a$. It
may be shown that the equation of the hyperbola is given by
$frac{y^2}{a^2} - frac{x^2}{b^2} = 1, where space c^2 = a^2 + b^2$
Hyperbolas have many useful applications, one of which is their use in
navigation systems to determine the location of a ship. Two
transmitting stations, with known positions transmit radio signals to
the ship. The difference in the reception times o
f the signals is used to
compute the difference in distance between the ship and the two transmitting stations. This
infomation places the ship on a hyperbola whose foci are the transmitting stations.
Suppose that radio stations are located at Tanga and Dar es Salaam, two cities on the
north-south coastline of Tanzania. Dar-es Salaam is located 200 km due south of Tanga (you
may assume that Dar es Salaam is directly south of Tanga). Simultaneous radio signals are
transmitted from Tanga and Dar es Salaam to a ship in the Indian Ocean. The ship receives
the signal from Tanga 500 microseconds (µs) before it receives the signal from Dar es Salaam.
Assume that the speed of radio signals is 300m/µs.
(a) By setting up an xy-coordinate system with Tanga having
coordinates (0, 100), determine the equation of the hyperbola on which
the ship lies.
(b) Given that the ship is due east of
Tanga, determine the coordinates of the ship.
If someone wouldnt mind giving me a few hints as to how I could solve this, I would be very grateful.
Thanks
Tim
conic-sections
asked Aug 15 '15 at 8:40
FishingfonFishingfon
3012718
$endgroup$
add a comment |
$begingroup$
I am currently studying multivariate calculus at university, and ive been given some practice problems before the first assignment.
The problem is:
A hyperbola may be defined as the set of points in a plane, the
difference of whose distances from two fixed points $F_1$ and $F_2$ is a
constant. Let P be a point on the hyperbola. Suppose the foci of the
hyperbola are located at (0, ±c), and that $|P F_1| − |P F_2| = ±2a$. It
may be shown that the equation of the hyperbola is given by
$frac{y^2}{a^2} - frac{x^2}{b^2} = 1, where space c^2 = a^2 + b^2$
Hyperbolas have many useful applications, one of which is their use in
navigation systems to determine the location of a ship. Two
transmitting stations, with known positions transmit radio signals to
the ship. The difference in the reception times o
f the signals is used to
compute the difference in distance between the ship and the two transmitting stations. This
infomation places the ship on a hyperbola whose foci are the transmitting stations.
Suppose that radio stations are located at Tanga and Dar es Salaam, two cities on the
north-south coastline of Tanzania. Dar-es Salaam is located 200 km due south of Tanga (you
may assume that Dar es Salaam is directly south of Tanga). Simultaneous radio signals are
transmitted from Tanga and Dar es Salaam to a ship in the Indian Ocean. The ship receives
the signal from Tanga 500 microseconds (µs) before it receives the signal from Dar es Salaam.
Assume that the speed of radio signals is 300m/µs.
(a) By setting up an xy-coordinate system with Tanga having
coordinates (0, 100), determine the equation of the hyperbola on which
the ship lies.
(b) Given that the ship is due east of
Tanga, determine the coordinates of the ship.
If someone wouldnt mind giving me a few hints as to how I could solve this, I would be very grateful.
Thanks
Tim
conic-sections
asked Aug 15 '15 at 8:40
FishingfonFishingfon
3012718
$endgroup$
I am currently studying multivariate calculus at university, and ive been given some practice problems before the first assignment.
The problem is:
A hyperbola may be defined as the set of points in a plane, the
difference of whose distances from two fixed points $F_1$ and $F_2$ is a
constant. Let P be a point on the hyperbola. Suppose the foci of the
hyperbola are located at (0, ±c), and that $|P F_1| − |P F_2| = ±2a$. It
may be shown that the equation of the hyperbola is given by
$frac{y^2}{a^2} - frac{x^2}{b^2} = 1, where space c^2 = a^2 + b^2$
Hyperbolas have many useful applications, one of which is their use in
navigation systems to determine the location of a ship. Two
transmitting stations, with known positions transmit radio signals to
the ship. The difference in the reception times o
f the signals is used to
compute the difference in distance between the ship and the two transmitting stations. This
infomation places the ship on a hyperbola whose foci are the transmitting stations.
Suppose that radio stations are located at Tanga and Dar es Salaam, two cities on the
north-south coastline of Tanzania. Dar-es Salaam is located 200 km due south of Tanga (you
may assume that Dar es Salaam is directly south of Tanga). Simultaneous radio signals are
transmitted from Tanga and Dar es Salaam to a ship in the Indian Ocean. The ship receives
the signal from Tanga 500 microseconds (µs) before it receives the signal from Dar es Salaam.
Assume that the speed of radio signals is 300m/µs.
(a) By setting up an xy-coordinate system with Tanga having
coordinates (0, 100), determine the equation of the hyperbola on which
the ship lies.
(b) Given that the ship is due east of
Tanga, determine the coordinates of the ship.
If someone wouldnt mind giving me a few hints as to how I could solve this, I would be very grateful.
Thanks
Tim
conic-sections
conic-sections
asked Aug 15 '15 at 8:40
FishingfonFishingfon
3012718
asked Aug 15 '15 at 8:40
FishingfonFishingfon
3012718
asked Aug 15 '15 at 8:40
FishingfonFishingfon
3012718
asked Aug 15 '15 at 8:40
FishingfonFishingfon
3012718
asked Aug 15 '15 at 8:40
FishingfonFishingfon
3012718
3012718
add a comment |
add a comment |
2 Answers
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It can be shown that the difference between the two distances (from ship to transmitting stations) is $2a$, where $a$ is the parameter appearing in the hyperbola equation. Let the distance between foci (that is between transmitting stations) be $2c$: you have then the relation $c^2=a^2+b^2$, whence you can compute $b$ and so determine the required equation. Of course you must also set the coordinate system so that Dar es Salaam has coordinates $(0,-100)$.
answered Aug 15 '15 at 9:20
AretinoAretino
25.9k31545
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Hey, Thanks for your reply, however I don't see how you compute b..? If you wouldn't mind explaining that it would be much appreciated. Thanks again, Tim
$endgroup$
– Fishingfon
Aug 15 '15 at 10:23
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@Fishingfon $b^2=c^2-a^2$, where $a$ and $c$ are computed from the data as I explained.
$endgroup$
– Aretino
Aug 15 '15 at 10:27
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Hey, Sorry Imeant I was not sure how you computed a, not b. I don't see how you compute a when we don know the distance from ship to transmitting station. Thanks again, Tim :)
$endgroup$
– Fishingfon
Aug 15 '15 at 10:33
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@Fishingfon Just think of the difference in the time signals are received, taking into account that signals travel at 300 m/µs.
$endgroup$
– Aretino
Aug 15 '15 at 10:47
add a comment |
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Apologies for the clumsily handwritten note. Hope it addresses the point.
answered Oct 8 '18 at 16:54
NarasimhamNarasimham
21.2k62258
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add a comment |
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2 Answers
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$begingroup$
It can be shown that the difference between the two distances (from ship to transmitting stations) is $2a$, where $a$ is the parameter appearing in the hyperbola equation. Let the distance between foci (that is between transmitting stations) be $2c$: you have then the relation $c^2=a^2+b^2$, whence you can compute $b$ and so determine the required equation. Of course you must also set the coordinate system so that Dar es Salaam has coordinates $(0,-100)$.
answered Aug 15 '15 at 9:20
AretinoAretino
25.9k31545
$endgroup$
$begingroup$
Hey, Thanks for your reply, however I don't see how you compute b..? If you wouldn't mind explaining that it would be much appreciated. Thanks again, Tim
$endgroup$
– Fishingfon
Aug 15 '15 at 10:23
$begingroup$
@Fishingfon $b^2=c^2-a^2$, where $a$ and $c$ are computed from the data as I explained.
$endgroup$
– Aretino
Aug 15 '15 at 10:27
$begingroup$
Hey, Sorry Imeant I was not sure how you computed a, not b. I don't see how you compute a when we don know the distance from ship to transmitting station. Thanks again, Tim :)
$endgroup$
– Fishingfon
Aug 15 '15 at 10:33
$begingroup$
@Fishingfon Just think of the difference in the time signals are received, taking into account that signals travel at 300 m/µs.
$endgroup$
– Aretino
Aug 15 '15 at 10:47
add a comment |
$begingroup$
It can be shown that the difference between the two distances (from ship to transmitting stations) is $2a$, where $a$ is the parameter appearing in the hyperbola equation. Let the distance between foci (that is between transmitting stations) be $2c$: you have then the relation $c^2=a^2+b^2$, whence you can compute $b$ and so determine the required equation. Of course you must also set the coordinate system so that Dar es Salaam has coordinates $(0,-100)$.
answered Aug 15 '15 at 9:20
AretinoAretino
25.9k31545
$endgroup$
$begingroup$
Hey, Thanks for your reply, however I don't see how you compute b..? If you wouldn't mind explaining that it would be much appreciated. Thanks again, Tim
$endgroup$
– Fishingfon
Aug 15 '15 at 10:23
$begingroup$
@Fishingfon $b^2=c^2-a^2$, where $a$ and $c$ are computed from the data as I explained.
$endgroup$
– Aretino
Aug 15 '15 at 10:27
$begingroup$
Hey, Sorry Imeant I was not sure how you computed a, not b. I don't see how you compute a when we don know the distance from ship to transmitting station. Thanks again, Tim :)
$endgroup$
– Fishingfon
Aug 15 '15 at 10:33
$begingroup$
@Fishingfon Just think of the difference in the time signals are received, taking into account that signals travel at 300 m/µs.
$endgroup$
– Aretino
Aug 15 '15 at 10:47
add a comment |
$begingroup$
It can be shown that the difference between the two distances (from ship to transmitting stations) is $2a$, where $a$ is the parameter appearing in the hyperbola equation. Let the distance between foci (that is between transmitting stations) be $2c$: you have then the relation $c^2=a^2+b^2$, whence you can compute $b$ and so determine the required equation. Of course you must also set the coordinate system so that Dar es Salaam has coordinates $(0,-100)$.
answered Aug 15 '15 at 9:20
AretinoAretino
25.9k31545
$endgroup$
It can be shown that the difference between the two distances (from ship to transmitting stations) is $2a$, where $a$ is the parameter appearing in the hyperbola equation. Let the distance between foci (that is between transmitting stations) be $2c$: you have then the relation $c^2=a^2+b^2$, whence you can compute $b$ and so determine the required equation. Of course you must also set the coordinate system so that Dar es Salaam has coordinates $(0,-100)$.
answered Aug 15 '15 at 9:20
AretinoAretino
25.9k31545
answered Aug 15 '15 at 9:20
AretinoAretino
25.9k31545
answered Aug 15 '15 at 9:20
AretinoAretino
25.9k31545
answered Aug 15 '15 at 9:20
AretinoAretino
25.9k31545
25.9k31545
$begingroup$
Hey, Thanks for your reply, however I don't see how you compute b..? If you wouldn't mind explaining that it would be much appreciated. Thanks again, Tim
$endgroup$
– Fishingfon
Aug 15 '15 at 10:23
$begingroup$
@Fishingfon $b^2=c^2-a^2$, where $a$ and $c$ are computed from the data as I explained.
$endgroup$
– Aretino
Aug 15 '15 at 10:27
$begingroup$
Hey, Sorry Imeant I was not sure how you computed a, not b. I don't see how you compute a when we don know the distance from ship to transmitting station. Thanks again, Tim :)
$endgroup$
– Fishingfon
Aug 15 '15 at 10:33
$begingroup$
@Fishingfon Just think of the difference in the time signals are received, taking into account that signals travel at 300 m/µs.
$endgroup$
– Aretino
Aug 15 '15 at 10:47
add a comment |
$begingroup$
Hey, Thanks for your reply, however I don't see how you compute b..? If you wouldn't mind explaining that it would be much appreciated. Thanks again, Tim
$endgroup$
– Fishingfon
Aug 15 '15 at 10:23
$begingroup$
@Fishingfon $b^2=c^2-a^2$, where $a$ and $c$ are computed from the data as I explained.
$endgroup$
– Aretino
Aug 15 '15 at 10:27
$begingroup$
Hey, Sorry Imeant I was not sure how you computed a, not b. I don't see how you compute a when we don know the distance from ship to transmitting station. Thanks again, Tim :)
$endgroup$
– Fishingfon
Aug 15 '15 at 10:33
$begingroup$
@Fishingfon Just think of the difference in the time signals are received, taking into account that signals travel at 300 m/µs.
$endgroup$
– Aretino
Aug 15 '15 at 10:47
$begingroup$
Hey, Thanks for your reply, however I don't see how you compute b..? If you wouldn't mind explaining that it would be much appreciated. Thanks again, Tim
$endgroup$
– Fishingfon
Aug 15 '15 at 10:23
$begingroup$
Hey, Thanks for your reply, however I don't see how you compute b..? If you wouldn't mind explaining that it would be much appreciated. Thanks again, Tim
$endgroup$
– Fishingfon
Aug 15 '15 at 10:23
$begingroup$
@Fishingfon $b^2=c^2-a^2$, where $a$ and $c$ are computed from the data as I explained.
$endgroup$
– Aretino
Aug 15 '15 at 10:27
$begingroup$
@Fishingfon $b^2=c^2-a^2$, where $a$ and $c$ are computed from the data as I explained.
$endgroup$
– Aretino
Aug 15 '15 at 10:27
$begingroup$
Hey, Sorry Imeant I was not sure how you computed a, not b. I don't see how you compute a when we don know the distance from ship to transmitting station. Thanks again, Tim :)
$endgroup$
– Fishingfon
Aug 15 '15 at 10:33
$begingroup$
Hey, Sorry Imeant I was not sure how you computed a, not b. I don't see how you compute a when we don know the distance from ship to transmitting station. Thanks again, Tim :)
$endgroup$
– Fishingfon
Aug 15 '15 at 10:33
$begingroup$
@Fishingfon Just think of the difference in the time signals are received, taking into account that signals travel at 300 m/µs.
$endgroup$
– Aretino
Aug 15 '15 at 10:47
$begingroup$
@Fishingfon Just think of the difference in the time signals are received, taking into account that signals travel at 300 m/µs.
$endgroup$
– Aretino
Aug 15 '15 at 10:47
add a comment |
$begingroup$
Apologies for the clumsily handwritten note. Hope it addresses the point.
answered Oct 8 '18 at 16:54
NarasimhamNarasimham
21.2k62258
$endgroup$
add a comment |
$begingroup$
Apologies for the clumsily handwritten note. Hope it addresses the point.
answered Oct 8 '18 at 16:54
NarasimhamNarasimham
21.2k62258
$endgroup$
add a comment |
$begingroup$
Apologies for the clumsily handwritten note. Hope it addresses the point.
answered Oct 8 '18 at 16:54
NarasimhamNarasimham
21.2k62258
$endgroup$
Apologies for the clumsily handwritten note. Hope it addresses the point.
answered Oct 8 '18 at 16:54
NarasimhamNarasimham
21.2k62258
answered Oct 8 '18 at 16:54
NarasimhamNarasimham
21.2k62258
answered Oct 8 '18 at 16:54
NarasimhamNarasimham
21.2k62258
answered Oct 8 '18 at 16:54
NarasimhamNarasimham
21.2k62258
21.2k62258
add a comment |
add a comment |
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