Question

identify the graph of the ellipse given by $\frac{(x + 7)^2}{49}+\frac{(y - 5)^2}{25}=1$

Explanation:

Step1: Find the center of the 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 given ellipse \(\frac{(x + 7)^{2}}{49}+\frac{(y - 5)^{2}}{25}=1\), we have \(h=-7\) and \(k = 5\). So the center is \((-7,5)\).

Step2: Analyze the major and minor axes

Since \(a^{2}=49\) (so \(a = 7\)) and \(b^{2}=25\) (so \(b = 5\)), and \(a>b\), the major axis is parallel to the \(x\) - axis.
The vertices are at \((h\pm a,k)\), so \((-7\pm7,5)\) which are \((0,5)\) and \((- 14,5)\).
The co - vertices are at \((h,k\pm b)\), so \((-7,5\pm5)\) which are \((-7,10)\) and \((-7,0)\).

By comparing the center \((-7,5)\) and the vertices and co - vertices with the given graphs, we can identify the correct graph.

Answer:

The first graph (the one with center near \((-7,5)\) and major axis parallel to the \(x\) - axis) is the correct graph of the ellipse \(\frac{(x + 7)^{2}}{49}+\frac{(y - 5)^{2}}{25}=1\)