Question

consider the following nonlinear system of equations.

$$ \\left\\{\
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\ ight. $$

step 2 of 3: determine the type and key parts of the graph of the second equation.
answer
selecting an option will display further inputs to complete your answer.
straight line center: (2,-1)
circle
ellipse length of major axis: 5
parabola
hyperbola length of minor axis: 1
major axis is:
horizontal vertical

Explanation:

Step1: Recall the standard form of an ellipse

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. For the 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 \(h = 2\), \(k=-1\), \(a = 5\), \(b = 1\).

Step2: Determine the center

The center of the ellipse \((h,k)\) is found by comparing with the standard form. So the center is \((2,-1)\).

Step3: Determine the lengths of major and minor axes

Since \(a>b\), the length of the major axis is \(2a\). Given \(a = 5\), the length of the major axis is \(2\times5=10\). The length of the minor axis is \(2b\). Given \(b = 1\), the length of the minor axis is \(2\times1 = 2\).

Step4: Determine the orientation of the major axis

Since the denominator of the \(x\) - term (\(a^{2}=25\)) is larger than the denominator of the \(y\) - term (\(b^{2} = 1\)), the major axis is horizontal.

Answer:

Type: ellipse; center: \((2,-1)\); length of major axis: \(10\); length of minor axis: \(2\); major axis is: horizontal