Question

The hyperbolic orbit of a comet is represented on a coordinate plane with center at (0, 4). One branch has a vertex at (0, 10) and its respective focus at (0, 14). Which equation represents the comet's orbit?

Answer 1

Answer: The final answer

Explanation:

The equation representing the hyperbolic orbit of a comet can be determined using the standard form of the equation for a hyperbola with a vertical transverse axis. The center is given as (0, 4), the vertex of one branch is (0, 10), and the focus is (0, 14). By plugging these values into the standard form equation and solving for b^2, we get the equation (y - 4)^2 / 36 - x^2 / (y - 14) = 1. Simplifying this equation gives the final answer, which is x^2 / 64 - (y - 4)^2 / 36 = 1, or the final answer. :)

Answer 2

Remember that

If the given coordinates of the vertices and foci have the form (0,10) and (0,14)

then

the transverse axis is the y-axis

so

the equation is of the form

(y-k)^2/a^2-(x-h)^2/b^2=1

In this problem

center (h,k) is equal to (0,4)

(0,a-k)) is equal to (0,10)

a=10-4=6

(0,c-k) is equal to (0,14)

c=14-4=10

Find out the value of b

b^2=c^2-a^2

b^2=10^2-6^2

b^2=64

therefore

the equation is equal to

(y-4)^2/36-x^2/64=1

the answer is option A