Answer:
A. There is a focus at (0,−10).
Step-by-step explanation:
Assume the hyperbola is like the one below.
The hyperbola is vertical and centred on the y-axis, so its general equation is
![\dfrac{y^{2}}{a^{2}} - \dfrac{x^{2}}{b^{2}} = 1](https://tex.z-dn.net/?f=%5Cdfrac%7By%5E%7B2%7D%7D%7Ba%5E%7B2%7D%7D%20-%20%5Cdfrac%7Bx%5E%7B2%7D%7D%7Bb%5E%7B2%7D%7D%20%3D%201)
The vertices of your parabola are (0,±8) so a = 8.
The covertices are (±6,0), so b = 6.
Calculate c
![\begin{array}{rcl}a^{2} + b^{2} & = & c^{2}\\8^{2} + 6^{2} & = & c^{2}\\64 + 36 & = & c^{2}\\100 & = & c^{2}\\c & = & \mathbf{10}\\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Brcl%7Da%5E%7B2%7D%20%2B%20b%5E%7B2%7D%20%26%20%3D%20%26%20c%5E%7B2%7D%5C%5C8%5E%7B2%7D%20%2B%206%5E%7B2%7D%20%26%20%3D%20%26%20c%5E%7B2%7D%5C%5C64%20%2B%2036%20%26%20%3D%20%26%20c%5E%7B2%7D%5C%5C100%20%26%20%3D%20%26%20c%5E%7B2%7D%5C%5Cc%20%26%20%3D%20%26%20%5Cmathbf%7B10%7D%5C%5C%5Cend%7Barray%7D)
A. Foci
The foci are at (0, ±c) = (0, ±10)
TRUE. There is a focus at (0,-10).
B. Foci
The foci are at (0,±10).
False. There is no focus at (0,12)
C. and D. Asymptotes
The equations for the asymptotes are
![y = \pm\dfrac{a}{b}x = \pm\dfrac{8}{6}x = \pm\dfrac{4}{3}x](https://tex.z-dn.net/?f=y%20%3D%20%5Cpm%5Cdfrac%7Ba%7D%7Bb%7Dx%20%3D%20%5Cpm%5Cdfrac%7B8%7D%7B6%7Dx%20%3D%20%5Cpm%5Cdfrac%7B4%7D%7B3%7Dx)
So, y = ±x are not asymptotes.
False.
E. and F. Directrices
The equations of the directrices are
y = ±a²/c = ±64/10 = ±6.4
y = 6.4 is a directrix.
E is false. x = cannot be a directrix
F is uncertain. Your equation for the directrix is incomplete.