Find the standard form of the equation of the ellipse satisfying the following conditions.
Vertices of major axis are ​(4​,12​) and ​(4​,6​)
The length of the minor axis is 4.
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Part 1
The standard form of the equation is the following.
  
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To find the standard form of the equation of the ellipse, let’s proceed step-by-step.

### Step 1: Identify key parameters  
The given conditions are:  
- **Vertices of the major axis**: \( (4, 12) \) and \( (4, 6) \)  
- **Length of the minor axis**: 4

### Step 2: Find the center of the ellipse  
The center is the **midpoint** of the major axis vertices:  
\[
\left( \frac{4 + 4}{2}, \frac{12 + 6}{2} \right) = (4, 9)
\]  
So, the **center** is \( (4, 9) \).

### Step 3: Determine the semi-major and semi-minor axes  
- **Distance between the vertices on the major axis**:  
\[
12 - 6 = 6
\]
Thus, the **length of the major axis** is 6, so the **semi-major axis** \(a = \frac{6}{2} = 3\).

- **Length of the minor axis**: Given as 4, so the **semi-minor axis** \(b = \frac{4}{2} = 2\).

### Step 4: Write the equation of the ellipse  
Since the **major axis is vertical**, the equation of the ellipse takes the form:  
\[
\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1
\]
where \( (h, k) \) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.

- \(h = 4\), \(k = 9\)
- \(a = 3\), so \(a^2 = 9\)
- \(b = 2\), so \(b^2 = 4\)

Thus, the equation becomes:
\[
\frac{(x - 4)^2}{4} + \frac{(y - 9)^2}{9} = 1
\]

### Final Answer  
The standard form of the equation of the ellipse is:
\[
\frac{(x - 4)^2}{4} + \frac{(y - 9)^2}{9} = 1
\]

Would you like further clarification or additional details?

### Related Questions  
1. How do you derive the formula for an ellipse?
2. How does the orientation of the major and minor axes affect the equation?
3. What is the general equation for ellipses with a horizontal major axis?
4. How do you calculate the foci of an ellipse?
5. What is the difference between ellipses and circles?

### Tip  
When the major axis is vertical, the larger denominator goes with the \(y\)-term; when it’s horizontal, the larger denominator is under the \(x\)-term.

Find the standard form of the equation of the ellipse satisfying the following conditions. Vertices of major axis are (4,12) and (4,6). The length of the minor axis is 4.

Learn how to find the standard form of the ellipse equation with vertices at (4,12) and (4,6), and a minor axis length of 4. This step-by-step guide covers key concepts like midpoint, distance formulas, and the equation of ellipses with vertical major axes. Suitable for Grades 10-12.

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