Find the equation of the ellipse with
focus at (0, 0), eccentricity is `5/6`, and directrix is `3x+4y-1=0`.

Answer

To find the equation of the ellipse with the given parameters, we will follow these steps:

### Step 1: Understand the parameters
We are given:
- Focus (S) at (0, 0)
- Eccentricity (e) = 5/6
- Directrix: 3x + 4y - 1 = 0

### Step 2: Write the formula for the ellipse
The definition of an ellipse states that for any point P(x, y) on the ellipse, the ratio of the distance from the focus to the distance from the directrix is equal to the eccentricity (e). This can be expressed as:
\[ \frac{PS}{PM} = e \]
where PS is the distance from the point P to the focus S, and PM is the distance from the point P to the directrix.

### Step 3: Calculate PS
The distance PS from point P(x, y) to the focus (0, 0) is given by:
\[ PS = \sqrt{x^2 + y^2} \]

### Step 4: Calculate PM
The distance PM from point P(x, y) to the directrix can be calculated using the formula for the distance from a point to a line. The directrix can be expressed in the form Ax + By + C = 0, where A = 3, B = 4, and C = -1. The distance PM is given by:
\[ PM = \frac{|3x + 4y - 1|}{\sqrt{3^2 + 4^2}} = \frac{|3x + 4y - 1|}{5} \]

### Step 5: Set up the equation
From the definition of the ellipse, we have:
\[ \frac{\sqrt{x^2 + y^2}}{\frac{|3x + 4y - 1|}{5}} = \frac{5}{6} \]
Cross-multiplying gives:
\[ 6\sqrt{x^2 + y^2} = 5 \cdot \frac{|3x + 4y - 1|}{5} \]
This simplifies to:
\[ 6\sqrt{x^2 + y^2} = |3x + 4y - 1| \]

### Step 6: Square both sides
To eliminate the square root, we square both sides:
\[ 36(x^2 + y^2) = (3x + 4y - 1)^2 \]

### Step 7: Expand the right-hand side
Expanding the right-hand side:
\[ (3x + 4y - 1)^2 = 9x^2 + 24xy + 16y^2 - 6x - 8y + 1 \]

### Step 8: Set the equation
Now we have:
\[ 36(x^2 + y^2) = 9x^2 + 24xy + 16y^2 - 6x - 8y + 1 \]
Rearranging gives:
\[ 36x^2 + 36y^2 - 9x^2 - 16y^2 - 24xy + 6x + 8y - 1 = 0 \]
This simplifies to:
\[ 27x^2 + 20y^2 - 24xy + 6x + 8y - 1 = 0 \]

### Final Equation
Thus, the equation of the ellipse is:
\[ 27x^2 + 20y^2 - 24xy + 6x + 8y - 1 = 0 \]

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