Question

the graph shows an ellipse. write its equation in standard form.

Explanation:

Step1: Identify the center

The center of the ellipse \( (h,k)\) can be observed from the graph. The center is at \((0, - 6)\), so \(h = 0\) and \(k=-6\).

Step2: Find the semi - major axis \(a\)

The distance from the center to the rightmost or leftmost point of the ellipse gives the semi - major axis. The center is at \((0,-6)\) and the rightmost point is at \((6,-6)\). So \(a=\vert6 - 0\vert=6\).

Step3: Find the semi - minor axis \(b\)

The distance from the center to the topmost or bottommost point of the ellipse gives the semi - minor axis. The center is at \((0,-6)\) and the bottommost point is at \((0,-8)\). So \(b=\vert-6-(-8)\vert = 2\).

Step4: Write the standard form of the ellipse equation

The standard form of an ellipse with a horizontal major axis is \(\frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}}=1\). Substituting \(h = 0\), \(k=-6\), \(a = 6\) and \(b = 2\) into the equation, we get \(\frac{x^{2}}{36}+\frac{(y + 6)^{2}}{4}=1\).

Answer:

\(\frac{x^{2}}{36}+\frac{(y + 6)^{2}}{4}=1\)