Question

1. find the center of the hyperbola \\( \frac{(x - 2)^{2}}{4^{2}}-\frac{(y - 1)^{2}}{2^{2}} = 1 \\).

center: \\( c(h,k)= \\)
\\( (2,-1) \\)
\\( (-2,1) \\)
\\( (2,1) \\)
\\( (-2,-1) \\)

Explanation:

Step1: Recall the standard form of a hyperbola

The standard form of a hyperbola is \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), where \((h,k)\) is the center of the hyperbola.

Step2: Identify \(h\) and \(k\)

For the given hyperbola \(\frac{(x - 2)^2}{4^2}-\frac{(y - 1)^2}{2^2}=1\), by comparing with the standard form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), we can see that \(h = 2\) and \(k=1\).

Answer:

\((2,1)\)