Answer:
The equation of the hyperbola can be presented as follows;
[tex]\dfrac{\left x \right ^{2}}{12} - \dfrac{\left y \right ^{2}}{24} = 1[/tex]
Step-by-step explanation:
We have the equation of an hyperbola given as follows;
[tex]\dfrac{\left (x - h \right )^{2}}{a^{2}} - \dfrac{\left (y - k \right )^{2}}{b^{2}} = 1[/tex]
The foal length = a² + b² = c²
c = focal length
The directrix = a²/c
The focal point are (h + c, k) and (h - c, k)
Therefore, by comparison with the given focal points, (6, 0) and (-6, 0), we have;
k = 0 and h + c = 6, h - c = -6
Therefore;
6 - c - c = -6
-2·c = -12
c = 6
h = 0
a²/c = 2
a² = 2 × 6 = 12
a = 2·√3
12 + b² = 6²
b² = 6² - 12 = 24
b² = 24
b = 2·√6
The equation of the hyper bola can then be written as follows;
[tex]\dfrac{\left (x - 0 \right )^{2}}{12} - \dfrac{\left (y - 0 \right )^{2}}{24} = 1[/tex]
Which gives
[tex]\dfrac{\left x \right ^{2}}{12} - \dfrac{\left y \right ^{2}}{24} = 1[/tex]
x² - y² = 12
y² = x² - 12.
