Question

find the equation of the following ellipse. express your answer in standard form.

Explanation:

Step1: Determine the center of the ellipse

The center of the ellipse \((h,k)\) is \((-1,-4)\) (by observing the position of the center dot on the coordinate - grid).

Step2: Find the values of \(a\) and \(b\)

The distance from the center to the right - hand (or left - hand) side of the ellipse (semi - major or semi - minor axis). The horizontal distance \(a\): The center is at \(x=-1\), and the ellipse extends from \(x = - 7\) to \(x = 5\). So \(a=\vert5-(-1)\vert=\vert-7 - (-1)\vert = 6\).
The vertical distance \(b\): The center is at \(y=-4\), and the ellipse extends from \(y=-1\) to \(y=-7\). So \(b=\vert-1-(-4)\vert=\vert-7-(-4)\vert = 3\).

Step3: Write the standard form of the ellipse equation

The standard form of an ellipse equation is \(\frac{(x - h)^{2}}{a^{2}}+\frac{(y - k)^{2}}{b^{2}}=1\) (since \(a>b\), the major axis is horizontal).
Substitute \(h=-1,k = - 4,a = 6,b = 3\) into the formula:

$$
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Answer:

\(\frac{(x + 1)^{2}}{36}+\frac{(y + 4)^{2}}{9}=1\)