The standard form equation of an ellipse is given by the formula:
For a vertical ellipse:
\( \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \
For a horizontal ellipse:
\( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \
In this case, the vertices of the ellipse are located at (0, 6) and (0, -6), indicating that the ellipse is vertical. The center of the ellipse, which is the midpoint of the line segment joining the vertices, is at (0, 0).
The distance from the center to each vertex (denoted by ‘a’) is 6, so \( a = 6 \). The co-vertices are at (4, 0) and (-4, 0), which shows that the distance from the center to each co-vertex (denoted by ‘b’) is 4, so \( b = 4 \).
Now, substituting these values into the standard form of the vertical ellipse, we have:
\( \frac{(x-0)^2}{4^2} + \frac{(y-0)^2}{6^2} = 1 \
which simplifies to:
\( \frac{x^2}{16} + \frac{y^2}{36} = 1 \
Thus, the standard form equation of the ellipse is:
\( \frac{x^2}{16} + \frac{y^2}{36} = 1 \