Find the standard form of an ellipse if the co vertices is (4, 8) and (10, 8), a focus at (3, 12) center with (7, 8) 

We are given the following information for an ellipse:

- Co-vertices: \( (4, 8) \) and \( (10, 8) \)
- One focus: \( (3, 12) \)
- Center: \( (7, 8) \)

### Step 1: Determine the center of the ellipse
The center of the ellipse is directly given as \( (7, 8) \).

### Step 2: Determine the distance from the center to the co-vertices
The co-vertices are at \( (4, 8) \) and \( (10, 8) \). These lie along the **minor axis** of the ellipse. The minor axis is horizontal because both points have the same \( y \)-coordinate, so the distance between the co-vertices is the length of the minor axis.

The distance between the center and a co-vertex is the semi-minor axis \( b \):
\[
b = \frac{10 - 4}{2} = 3
\]
Thus, the length of the semi-minor axis is \( b = 3 \).

### Step 3: Determine the distance from the center to the focus
The focus lies at \( (3, 12) \). The distance between the center \( (7, 8) \) and the focus is the distance along the major axis, which we can calculate using the distance formula:
\[
c = \sqrt{(7 - 3)^2 + (8 - 12)^2} = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]
Thus, \( c = 4\sqrt{2} \).

### Step 4: Use the relationship between \( a \), \( b \), and \( c \)
In an ellipse, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the focal distance \( c \) is given by:
\[
c^2 = a^2 - b^2
\]
Substitute \( c = 4\sqrt{2} \) and \( b = 3 \) into this equation:
\[
(4\sqrt{2})^2 = a^2 - 3^2
\]
\[
32 = a^2 - 9
\]
\[
a^2 = 41
\]
Thus, \( a = \sqrt{41} \).

### Step 5: Write the standard form of the equation
The standard form of the equation of an ellipse is:
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
\]
where \( (h, k) \) is the center, \( a \) is the length of the semi-major axis, and \( b \) is the length of the semi-minor axis.

Substitute \( h = 7 \), \( k = 8 \), \( a^2 = 41 \), and \( b^2 = 9 \):
\[
\frac{(x - 7)^2}{41} + \frac{(y - 8)^2}{9} = 1
\]

Thus, the equation of the ellipse in standard form is:
\[
\boxed{\frac{(x - 7)^2}{41} + \frac{(y - 8)^2}{9} = 1}
\]

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:
1. How do you find the foci of an ellipse if the equation is given?
2. What is the geometric interpretation of \( a \), \( b \), and \( c \) in an ellipse?
3. How do you identify the major and minor axes from an ellipse's equation?
4. Can an ellipse have its major axis along the \( y \)-axis instead of the \( x \)-axis?
5. What are the key differences between an ellipse and a hyperbola?

**Tip:** Always verify the center and orientation of an ellipse from its co-vertices and foci before writing the equation.

Find the standard form of an ellipse if the co-vertices is (4, 8) and (10, 8), a focus at (3, 12) center with (7, 8)

Learn how to find the standard form of an ellipse given its co-vertices at (4, 8) and (10, 8), focus at (3, 12), and center at (7, 8). This problem covers key concepts in conic sections, including the ellipse equation and the relationship between the axes and foci. Suitable for high school students in Grades 10-12.

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