A hyperbola is centered at C = (-6,8) opens with vertices at (-6,11) and (-6,5), and the slopes of the asymptotes are m = +- 3/7.
The equation of the hyperbola is:
a) (x+6)^2 - (y-8)^2 = 1
b) (y-8)^2 - (x+6)^2 = 1
Find the missing denominators.
The given hyperbola has its center at C = (-6, 8) and opens with vertices at (-6, 11) and (-6, 5) respectively. Since the center, and the vertices all have changing y-coordinate, the hyperbola is one with its transverse axis vertical.
Therefore, equation must have negative x-term, i.e. the one similar to the equation in option b).
The vertices of a hyperbola are placed ‘a’ distance away from its centre. With center at c=(-6, 8) and changing y-coordinates, the coordinates of its vertices would be at (-6, 8+-a).
Comparing its given vertices at (-6, 11) and (-6, 5), we get,
and again, `8-a=5`
`rArr a=3` .
The slope of the asymptotes is given by `+-a/b` (transverse axis, vertical)
Comparing the given values yield `a/b=3/7` putting the value of a,
Therefore, the required equation of the hyperbola is
Hence the denominators of the two terms in equation b) are 9 and 49 respectively.