Question

consider the following hyperbola.
$$ \frac { ( y - 5 ) ^ { 2 } } { 81 } - \frac { ( x + 7 ) ^ { 2 } } { 4 } = 1 $$
step 1 of 3: find the coordinates of the vertices of the hyperbola.

Explanation:

Step1: Identify the standard form of the hyperbola

The standard form of a vertical hyperbola is \(\frac{(y - k)^2}{a^2}-\frac{(x - h)^2}{b^2}=1\), where \((h,k)\) is the center of the hyperbola.
For the given hyperbola \(\frac{(y - 5)^2}{81}-\frac{(x + 7)^2}{4}=1\), we have \(h=-7\), \(k = 5\), \(a^2=81\) (so \(a = 9\)), \(b^2 = 4\) (so \(b=2\)).

Step2: Find the coordinates of the vertices

The vertices of a vertical hyperbola \(\frac{(y - k)^2}{a^2}-\frac{(x - h)^2}{b^2}=1\) are given by the formula \((h,k\pm a)\).
Substitute \(h=-7\), \(k = 5\), and \(a = 9\) into the formula:
For the first vertex: \(y=k + a=5+9 = 14\), \(x=h=-7\), so the vertex is \((-7,14)\).
For the second vertex: \(y=k - a=5-9=-4\), \(x=h=-7\), so the vertex is \((-7,-4)\).

Answer:

The coordinates of the vertices are \((-7,14)\) and \((-7,-4)\).