Question

question 1
give the equation for the ellipse graphed above.

Explanation:

Step1: Identify the center of the ellipse

The center of the ellipse is at the origin \((0,0)\) since the graph is symmetric about both the \(x\)-axis and \(y\)-axis, and the midpoint of the vertices and co - vertices is \((0,0)\).

Step2: Determine the major and minor axes

• For the \(y\) - axis: The ellipse goes from \(y = - 10\) to \(y=10\), so the length of the semi - major axis (along the \(y\) - axis) \(a\) is the distance from the center \((0,0)\) to the vertex \((0,10)\) or \((0, - 10)\). So \(a = 10\).
• For the \(x\) - axis: The ellipse goes from \(x=-2\) to \(x = 2\), so the length of the semi - minor axis (along the \(x\) - axis) \(b\) is the distance from the center \((0,0)\) to the co - vertex \((2,0)\) or \((-2,0)\). So \(b = 2\).

Step3: Write the equation of the ellipse

The standard form of the equation of an ellipse with center \((h,k)=(0,0)\), major axis along the \(y\) - axis is \(\frac{(x - h)^2}{b^2}+\frac{(y - k)^2}{a^2}=1\).

Substituting \(h = 0\), \(k = 0\), \(a = 10\), and \(b = 2\) into the formula, we get \(\frac{x^{2}}{2^{2}}+\frac{y^{2}}{10^{2}}=1\), which simplifies to \(\frac{x^{2}}{4}+\frac{y^{2}}{100}=1\).

Answer:

\(\frac{x^{2}}{4}+\frac{y^{2}}{100}=1\)