Question

answer the following questions.

1. using the equation \\( \frac { ( x - 2 ) ^ { 2 } } { 4 ^ { 2 } } - \frac { ( y - 1 ) ^ { 2 } } { 2 ^ { 2 } } = 1 \\), find \\( a, b, h \\) and \\( k \\).

\\( a = \\)
\\( b = \\)
\\( h = \\)
\\( k = \\)

Explanation:

Step1: Recall the standard form of hyperbola

The standard form of a hyperbola \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}}=1\) (opens left - right).

Step2: Compare with the given equation

Given equation \(\frac{(x - 2)^{2}}{4^{2}}-\frac{(y - 1)^{2}}{2^{2}}=1\).
By comparing \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}}=1\) and \(\frac{(x - 2)^{2}}{4^{2}}-\frac{(y - 1)^{2}}{2^{2}}=1\):

• For \(a\): Since \(a^{2}=4^{2}\), then \(a = 4\).
• For \(b\): Since \(b^{2}=2^{2}\), then \(b = 2\).
• For \(h\): Comparing \((x - h)\) and \((x - 2)\), we get \(h=2\).
• For \(k\): Comparing \((y - k)\) and \((y - 1)\), we get \(k = 1\).

Answer:

\(a = 4\), \(b=2\), \(h = 2\), \(k=1\)