Find the equation to the ellipse with axes as the axes of coordinates.
the minor axis is equal to the distance between the foci, and the latus rectum is 10.

Answer

To find the equation of the ellipse given the conditions, we can follow these steps:

### Step 1: Understand the properties of the ellipse
The ellipse has its major axis along the x-axis and its minor axis along the y-axis. The standard form of the equation of such an ellipse is given by:

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

where \(2a\) is the length of the major axis and \(2b\) is the length of the minor axis.

### Step 2: Identify the relationship between the minor axis and the distance between the foci
The distance between the foci \(F_1\) and \(F_2\) of the ellipse is given by \(2ae\), where \(e\) is the eccentricity of the ellipse. The eccentricity \(e\) can be expressed as:

\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

From the problem, we know that the minor axis \(2b\) is equal to the distance between the foci \(2ae\). Therefore, we can write:

\[ 2b = 2ae \implies b = ae \]

### Step 3: Use the latus rectum condition
The latus rectum \(L\) of the ellipse is given by:

\[ L = \frac{2b^2}{a} \]

From the problem, we know that the latus rectum is equal to 10:

\[ \frac{2b^2}{a} = 10 \implies 2b^2 = 10a \implies b^2 = 5a \]

### Step 4: Substitute \(b\) from the first equation into the second equation
We have two equations now:
1. \(b = ae\)
2. \(b^2 = 5a\)

Substituting \(b = ae\) into \(b^2 = 5a\):

\[ (ae)^2 = 5a \implies a^2e^2 = 5a \]

Assuming \(a \neq 0\), we can divide both sides by \(a\):

\[ ae^2 = 5 \implies e^2 = \frac{5}{a} \]

### Step 5: Substitute \(e^2\) back into the eccentricity formula
We know that:

\[ e^2 = 1 - \frac{b^2}{a^2} \]

Substituting \(b^2 = 5a\):

\[ \frac{5}{a} = 1 - \frac{5a}{a^2} \implies \frac{5}{a} = 1 - \frac{5}{a} \implies \frac{10}{a} = 1 \implies a = 10 \]

### Step 6: Find \(b\)
Now substituting \(a = 10\) back into \(b^2 = 5a\):

\[ b^2 = 5 \times 10 = 50 \implies b = \sqrt{50} = 5\sqrt{2} \]

### Step 7: Write the equation of the ellipse
Now we can substitute \(a\) and \(b\) into the standard form of the ellipse:

\[ \frac{x^2}{10^2} + \frac{y^2}{(5\sqrt{2})^2} = 1 \]

This simplifies to:

\[ \frac{x^2}{100} + \frac{y^2}{50} = 1 \]

### Final Equation
Thus, the equation of the ellipse is:

\[ \frac{x^2}{100} + \frac{y^2}{50} = 1 \]