Question

The hyperbolic orbit of a comet is represented on a coordinate plane with center at (0, 7). One branch has a vertex at (0, 4) and its respective focus at (0, 2). Which equation represents the comet's orbit?A. x^2/16 - (y-7)^2/9 =1B. x^2/25 - (y-7)^2/9 =1C. (y-7)^2/25 - x^2/9=1D. (y-7)^2/9 - x^2/16=1

Answer 1

From the given information, we can to note that the hyperbola is vertical, that is, it opens upwards (and downward for the other branch). Then, the possible solutions are option C and D and it has the form

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

where (h,k) corresponds to the center coordinates and a represents the distance from the center to the vertex. Then, from the given information, a is given by

`a=7-4=3`

So, our hyperbola has the form

`\begin{gathered} ((y-7)^2)/(3^2)-((x-0)^2)/(b^2)=1 \\ \end{gathered}`

that is,

`((y-7)^2)/(9)-(x^2)/(b^2)=1`

So by comparing this last result with the given options, we can to note that options D has the same "a" squared ( which is 9). Therefore, the answer is option D