Question

1. using the equation \\( \frac { ( x - 2 ) ^ { 2 } } { 9 } - \frac { ( y - 1 ) ^ { 2 } } { 4 } = 1 \\) find the vertices. \\( ( - 5,1 ) \\) \\( ( 5,1 ) \\) \\( ( - 1,1 ) \\) \\( ( - 5, - 1 ) \\) \\( ( 1, - 1 ) \\)

Explanation:

Step1: Recall the standard form of hyperbola

The standard form of a hyperbola \(\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}}=1\) (horizontal transverse axis) has center \((h,k)\), vertices \((h\pm a,k)\)

Step2: Identify \(h,k,a\) from the given equation

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

Step3: Calculate the vertices

Using the formula \((h\pm a,k)\), we get \((2 + 3,1)=(5,1)\) and \((2-3,1)=(-1,1)\)

Answer:

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