Question

MEDAL AND POINTS!!!!! write equation: The hyperbola with center (1, 2), vertices (1, 5) and (1, –1) , and foci (1, 7) and (1, –3).

Answer 1

We notice the x of the foci don't change
that means the hyperbola opens up and down
$\frac{(y-k)^2}{a^2}- \frac{(x-h)^2}{b^2}=1$
hmm
center is (1,2) and vertices (1,5) and (1,-1)
the distance to a vertex is 3, that is 'a'
the rectangle of refernce is 6 units high
remember that
a²+b²=c² where a is major axis, b is minor axis and c is the distance from focs to center
from focus to center is 5 units
a is 3
3²+b²=5²
9+b²=25
mnus 9
b²=16
sqrt
b=4


so
$\frac{(y-2)^2}{3^2}- \frac{(x-1)^2}{4^2}=1$ is the equation