Question

consider the following nonlinear system of equations.

$$ left{ \begin{array} { l } { ( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 36 } \\ { \frac { ( x - 2 ) ^ { 2 } } { 25 } + ( y + 1 ) ^ { 2 } = 1 } end{array} ight. $$

step 2 of 3: determine the type and key parts of the graph of the second equation.
answer
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straight line
circle
ellipse
parabola
hyperbola

Explanation:

Step1: Recall the standard form of conic sections

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

Step2: Analyze the given equation \(\frac{(x - 2)^{2}}{25}+(y + 1)^{2}=1\)

We can rewrite it as \(\frac{(x - 2)^{2}}{5^{2}}+\frac{(y+1)^{2}}{1^{2}} = 1\). Here, \(h = 2\), \(k=-1\), \(a = 5\), \(b = 1\).

Answer:

ellipse with center \((2,-1)\), semi - major axis \(a = 5\), semi - minor axis \(b = 1\)