Question

which of the following is the graph of ( 16 x ^ { 2 } + 25 y ^ { 2 } = 400 )?

Explanation:

Step1: Simplify the equation

Divide both sides of \(16x^{2}+25y^{2}=400\) by \(400\).
We get \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\). This is the standard form of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1\) (\(a>b>0\)), where \(a = 5\) (semi - major axis along the \(x\) - axis) and \(b = 4\) (semi - minor axis along the \(y\) - axis).

Step2: Analyze the properties of the ellipse

For the ellipse \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\), the vertices on the \(x\) - axis are \((\pm5,0)\) and the vertices on the \(y\) - axis are \((0,\pm4)\).

Answer:

The graph of the equation \(16x^{2}+25y^{2}=400\) (or \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)) is an ellipse centered at the origin with semi - major axis \(a = 5\) (along the \(x\) - axis) and semi - minor axis \(b=4\) (along the \(y\) - axis). So, among the given options (assuming the second option has vertices at \((\pm5,0)\) and \((0,\pm4)\) and is an ellipse centered at the origin), the second graph (the one that is an ellipse with a wider span along the \(x\) - axis) is the correct graph of \(16x^{2}+25y^{2}=400\).