Find the equation to the ellipse with axes as the axes of coordinates.
distance of the focus from the corresponding directrix is 9 and eccentricity is `4/5`,

Answer

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

### Step 1: Understand the properties of the ellipse
The ellipse has its axes aligned with the coordinate axes, which means its standard form is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
where \(a\) is the semi-major axis and \(b\) is the semi-minor axis.

### Step 2: Use the given information
We know:
- The distance from the focus to the corresponding directrix is \(9\).
- The eccentricity \(e\) is given as \(\frac{4}{5}\).

### Step 3: Relate the focus and directrix
For an ellipse, the coordinates of the foci are \((\pm ae, 0)\) and the equations of the directrices are \(x = \pm \frac{a}{e}\).

### Step 4: Set up the distance equation
The distance \(PQ\) between the focus and the directrix can be expressed as:
\[ PQ = \frac{a}{e} - ae \]
Given that \(PQ = 9\), we can write:
\[ \frac{a}{e} - ae = 9 \]

### Step 5: Substitute the value of \(e\)
Substituting \(e = \frac{4}{5}\):
\[ \frac{a}{\frac{4}{5}} - a \cdot \frac{4}{5} = 9 \]
This simplifies to:
\[ \frac{5a}{4} - \frac{4a}{5} = 9 \]

### Step 6: Find a common denominator
The common denominator for \(4\) and \(5\) is \(20\):
\[ \frac{5a \cdot 5}{20} - \frac{4a \cdot 4}{20} = 9 \]
This simplifies to:
\[ \frac{25a - 16a}{20} = 9 \]
\[ \frac{9a}{20} = 9 \]

### Step 7: Solve for \(a\)
Multiplying both sides by \(20\):
\[ 9a = 180 \]
Dividing by \(9\):
\[ a = 20 \]

### Step 8: Find \(b\) using the relationship between \(a\), \(b\), and \(e\)
We use the relationship:
\[ b^2 = a^2(1 - e^2) \]
Calculating \(e^2\):
\[ e^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]
Now substituting \(a = 20\):
\[ b^2 = 20^2 \left(1 - \frac{16}{25}\right) = 400 \left(\frac{9}{25}\right) \]
\[ b^2 = \frac{3600}{25} = 144 \]

### Step 9: Write the equation of the ellipse
Now that we have \(a^2 = 400\) and \(b^2 = 144\), we can write the equation of the ellipse:
\[ \frac{x^2}{400} + \frac{y^2}{144} = 1 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ \frac{x^2}{400} + \frac{y^2}{144} = 1 \]