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A comet follows a hyperbolic path in which the sun is located at one of its foci.
The path of the comet can be modeled by the hyperbola shown below,
centered at the origin and opening left/right.
If the closest distance the comet reaches to the sun is 160 million km,
and the sun is 325 million km from the center of the hyperbola,
write an equation of the hyperbola (in millions of km).
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We are given that the hyperbola is centered at the origin of the coordinar system
and is opened left/right. It means that the hyperbola equation has the form
- = 1
with unknown now coefficients "a" and "b".
As given, the sun is in one of the focuses on x-axis, let say in the focus (c,0) = (325,0),
where c = 325 represents the distance of 325 millions kilometers from the origin to the sun.
So c= 325 is half the focal distance of this hyperbola.
Next, since the closest distance the comet reaches to the sun is 160 million km (given),
it implies that the vertex of the hyperbola is the point x = 325 - 160 = 165.
Hence, in equation (1) the coeddicient "a" is 165, representing 165 million kilometers
from the origin to the hyperbola vertex.
Now, knowving "a" and "c", we can find the coefficient b
b = = = = 280.
Thus, the equation of the hypebola, in million kilometers, is
- = 1.
In the solution, 280 millions kilometers is half the length of the imaginary axis of the hyperbola.
Solved.
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For the canonical equation of hyperbola, see the lesson
- Hyperbola definition, canonical equation, characteristic points and elements
in this site.
The comets that have hyperbolic orbits are those that come to " vicinity " of the sun ONLY ONCE.
They have too big velocity in order for the Sun's gravity be able to return them to its vicinity again.
For the list of such comets, see this Wikipedia web-page
https://en.wikipedia.org/wiki/List_of_hyperbolic_comets#:~:text=By%20definition%2C%20a%20hyperbolic%20orbit,its%20eccentricity%20is%20otherwise%20changed.