Find the eqation of the ellipse whose co-ordinates of focus are (3,2), eccentricity is `(2)/(3)` and equation of directrix is 3x+4y+5=0.

Answer

To find the equation of the ellipse with the given focus, eccentricity, and directrix, we can follow these steps:

### Step 1: Identify Given Values
- Focus (F) = (3, 2)
- Eccentricity (e) = \( \frac{2}{3} \)
- Equation of Directrix: \( 3x + 4y + 5 = 0 \)

### Step 2: Determine the Distance from a Point to the Focus
Let \( P(x, y) \) be a point on the ellipse. The distance from point \( P \) to the focus \( F(3, 2) \) is given by:
\[ PF = \sqrt{(x - 3)^2 + (y - 2)^2} \]

### Step 3: Determine the Distance from a Point to the Directrix
The distance from point \( P(x, y) \) to the directrix \( 3x + 4y + 5 = 0 \) can be calculated using the formula for the distance from a point to a line:
\[ PN = \frac{|3x + 4y + 5|}{\sqrt{3^2 + 4^2}} = \frac{|3x + 4y + 5|}{5} \]

### Step 4: Set Up the Relationship for the Ellipse
According to the definition of an ellipse, the distance from the point \( P \) to the focus \( F \) is equal to the eccentricity \( e \) times the distance from \( P \) to the directrix:
\[ PF = e \cdot PN \]
Substituting the distances we found:
\[ \sqrt{(x - 3)^2 + (y - 2)^2} = \frac{2}{3} \cdot \frac{|3x + 4y + 5|}{5} \]

### Step 5: Square Both Sides to Eliminate the Square Root
Squaring both sides gives:
\[ (x - 3)^2 + (y - 2)^2 = \left(\frac{2}{3} \cdot \frac{|3x + 4y + 5|}{5}\right)^2 \]
This simplifies to:
\[ (x - 3)^2 + (y - 2)^2 = \frac{4}{9} \cdot \frac{(3x + 4y + 5)^2}{25} \]
\[ (x - 3)^2 + (y - 2)^2 = \frac{4(3x + 4y + 5)^2}{225} \]

### Step 6: Clear the Denominator
Multiply both sides by 225 to eliminate the fraction:
\[ 225((x - 3)^2 + (y - 2)^2) = 4(3x + 4y + 5)^2 \]

### Step 7: Expand Both Sides
Expand the left side:
\[ 225((x^2 - 6x + 9) + (y^2 - 4y + 4)) = 225(x^2 + y^2 - 6x - 4y + 13) \]
\[ = 225x^2 + 225y^2 - 1350x - 900y + 2925 \]

Expand the right side:
\[ 4(3x + 4y + 5)^2 = 4(9x^2 + 24xy + 16y^2 + 30x + 40y + 25) \]
\[ = 36x^2 + 96xy + 64y^2 + 120x + 160y + 100 \]

### Step 8: Rearrange the Equation
Set the equation to zero:
\[ 225x^2 + 225y^2 - 1350x - 900y + 2925 - (36x^2 + 96xy + 64y^2 + 120x + 160y + 100) = 0 \]
Combine like terms:
\[ (225 - 36)x^2 + (225 - 64)y^2 - 96xy - (1350 + 120)x - (900 + 160)y + (2925 - 100) = 0 \]
\[ 189x^2 + 161y^2 - 96xy - 1470x - 1060y + 2825 = 0 \]

### Final Equation of the Ellipse
Thus, the equation of the ellipse is:
\[ 189x^2 + 161y^2 - 96xy - 1470x - 1060y + 2825 = 0 \]