Question

1. using the equation $\frac{(x - 2)^2}{9}-\frac{(y - 1)^2}{4}=1$ find the vertices. select all that apply.

$(h + a,k)$ and $(h - a,k)$
$(1,5)$
$(-1,-1)$
$(1,-1)$
$(5,1)$
$(-1,1)$

Explanation:

Step1: Identify \(h\), \(k\), \(a\)

For the hyperbola equation \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}} = 1\), comparing with \(\frac{(x - 2)^{2}}{9}-\frac{(y - 1)^{2}}{4}=1\), we have \(h = 2\), \(k = 1\), \(a^{2}=9\) (so \(a = 3\)).

Step2: Calculate vertices

Using the formula \((h + a,k)\) and \((h - a,k)\):

• For \((h + a,k)\): Substitute \(h = 2\), \(a = 3\), \(k = 1\) into \((h + a,k)\), we get \((2+3,1)=(5,1)\).
• For \((h - a,k)\): Substitute \(h = 2\), \(a = 3\), \(k = 1\) into \((h - a,k)\), we get \((2 - 3,1)=(-1,1)\).

Answer:

\((5,1)\), \((-1,1)\)