Question

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

asymptotes: \\( y = \pm\frac{b}{a}(x - h)+k=\\)__,__
\\( \square y=-\frac{1}{2}x \\)
\\( \square y=\frac{1}{2}x - 2 \\)
\\( \square y=\frac{1}{2}x + 2 \\)
\\( \square y=\frac{1}{2}x \\)
\\( \square y=-\frac{1}{2}x + 2 \\)

Explanation:

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

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

Step2: Substitute into the asymptote formula

Substitute into \(y=\pm\frac{b}{a}(x - h)+k\).
For the positive slope:
\(y=\frac{2}{4}(x - 2)+1=\frac{1}{2}x-1 + 1=\frac{1}{2}x\)
For the negative slope:
\(y=-\frac{2}{4}(x - 2)+1=-\frac{1}{2}x + 1+1=-\frac{1}{2}x+2\)

Answer:

\(y = \frac{1}{2}x\), \(y=-\frac{1}{2}x + 2\)