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Find the standard equation of thd given ellipse 

To find the standard equation of the given ellipse, we first need to extract the key parameters from the image:

### 1. **Center of the Ellipse**:
- From the graph, the center of the ellipse appears to be at the origin, \((0, 0)\).

### 2. **Axes of the Ellipse**:
- The major axis (longer axis) seems to be vertical, extending from \(y = 6\) to \(y = -6\), so its length is \(12\) units.
- The minor axis (shorter axis) extends horizontally from \(x = -4\) to \(x = 4\), making its length \(8\) units.

### 3. **Semi-major and Semi-minor Axes**:
- The semi-major axis (half the length of the major axis) is \(6\) units.
- The semi-minor axis (half the length of the minor axis) is \(4\) units.

### 4. **Standard Equation**:
The standard form of the equation of an ellipse with a vertical major axis and center at the origin is:

\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]

Where:
- \(a = 6\) is the semi-major axis.
- \(b = 4\) is the semi-minor axis.

Substituting these values into the formula gives:

\[
\frac{x^2}{4^2} + \frac{y^2}{6^2} = 1
\]

Simplifying:

\[
\frac{x^2}{16} + \frac{y^2}{36} = 1
\]

So, the standard equation of the given ellipse is:

\[
\boxed{\frac{x^2}{16} + \frac{y^2}{36} = 1}
\]

Would you like any further details or clarifications?

### Related Questions:
1. How do you derive the equation of an ellipse when its center is not at the origin?
2. What happens to the equation if the major axis is horizontal instead of vertical?
3. Can you explain the difference between an ellipse and a circle in terms of their equations?
4. How do you calculate the foci of an ellipse given its semi-major and semi-minor axes?
5. What is the general form of an ellipse's equation with center \((h, k)\)?

### Tip:
To distinguish between a vertical and horizontal major axis, always check which axis (x or y) has the larger denominator in the standard equation.

Find the standard equation of the given ellipse.

Discover how to find the standard equation of an ellipse with a vertical major axis, given its center at the origin, with semi-major and semi-minor axes. This detailed solution provides a step-by-step approach to solving the problem for students in Grades 10-12.

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