Question
Find an equation that models the paths of a satellite if its path is a hyperbola, a=45,000 km, and c=71,000 km. Assume that the center of the hyperbola is the origin and the transverse axis is horizontal.
Answers
Since the hyperbola is centered at the origin and the transverse axis is horizontal, the equation has the form:
x^2/a^2 – y^2/b^2 = 1
where a is the distance from the origin to either vertex, and b is the distance from the origin to either foci. We are given that a = 45,000 km, so:
x^2/45,000^2 – y^2/b^2 = 1
To find b, we use the relationship c^2 = a^2 + b^2, where c is the distance from the origin to either focus. We are given that c = 71,000 km, so:
71,000^2 = 45,000^2 + b^2
b^2 = 71,000^2 – 45,000^2
b^2 = 2,696,000,000 km^2
Now we can substitute this value of b^2 into the equation:
x^2/45,000^2 – y^2/2,696,000,000 = 1
This equation models the paths of the satellite if its path is a hyperbola, with a distance of 45,000 km from the origin to either vertex and 71,000 km from the origin to either focus.
x^2/a^2 – y^2/b^2 = 1
where a is the distance from the origin to either vertex, and b is the distance from the origin to either foci. We are given that a = 45,000 km, so:
x^2/45,000^2 – y^2/b^2 = 1
To find b, we use the relationship c^2 = a^2 + b^2, where c is the distance from the origin to either focus. We are given that c = 71,000 km, so:
71,000^2 = 45,000^2 + b^2
b^2 = 71,000^2 – 45,000^2
b^2 = 2,696,000,000 km^2
Now we can substitute this value of b^2 into the equation:
x^2/45,000^2 – y^2/2,696,000,000 = 1
This equation models the paths of the satellite if its path is a hyperbola, with a distance of 45,000 km from the origin to either vertex and 71,000 km from the origin to either focus.