Find the equation of the ellipse whose co-ordinates of focus are (1,2), eccentricity is `(1)/(3)` and equation of directrix is x+5y=6.

Answer

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

### Step 1: Identify the given parameters
- Focus (F) = (1, 2)
- Eccentricity (e) = \( \frac{1}{3} \)
- Equation of directrix = \( x + 5y - 6 = 0 \)

### Step 2: Define the point P on the ellipse
Let P be a point on the ellipse with coordinates (x, y).

### Step 3: Calculate the distance PF (from point P to focus F)
Using the distance formula, we have:
\[ PF = \sqrt{(x - 1)^2 + (y - 2)^2} \]

### Step 4: Calculate the distance PN (from point P to the directrix)
The distance from point P to the directrix can be calculated using the formula:
\[ PN = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
For the directrix \( x + 5y - 6 = 0 \), we have:
- A = 1, B = 5, C = -6
Thus,
\[ PN = \frac{|1 \cdot x + 5 \cdot y - 6|}{\sqrt{1^2 + 5^2}} = \frac{|x + 5y - 6|}{\sqrt{26}} \]

### Step 5: Set up the relationship between PF and PN
According to the definition of an ellipse:
\[ PF = e \cdot PN \]
Substituting the values we have:
\[ \sqrt{(x - 1)^2 + (y - 2)^2} = \frac{1}{3} \cdot \frac{|x + 5y - 6|}{\sqrt{26}} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (x - 1)^2 + (y - 2)^2 = \left(\frac{1}{3} \cdot \frac{|x + 5y - 6|}{\sqrt{26}}\right)^2 \]
This simplifies to:
\[ (x - 1)^2 + (y - 2)^2 = \frac{1}{9} \cdot \frac{(x + 5y - 6)^2}{26} \]

### Step 7: Multiply through by 234 to eliminate fractions
Multiply both sides by 234 (which is \( 9 \times 26 \)):
\[ 234 \left((x - 1)^2 + (y - 2)^2\right) = (x + 5y - 6)^2 \]

### Step 8: Expand both sides
Expanding the left-hand side:
\[ 234 \left((x^2 - 2x + 1) + (y^2 - 4y + 4)\right) = 234x^2 - 468x + 234 + 234y^2 - 936y + 936 \]
This simplifies to:
\[ 234x^2 + 234y^2 - 468x - 936y + 1170 \]

Expanding the right-hand side:
\[ (x + 5y - 6)^2 = x^2 + 10xy + 25y^2 - 12x - 60y + 36 \]

### Step 9: Set the equation to zero
Combine both sides:
\[ 234x^2 + 234y^2 - 468x - 936y + 1170 - (x^2 + 10xy + 25y^2 - 12x - 60y + 36) = 0 \]
This leads to:
\[ (234 - 1)x^2 + (234 - 25)y^2 - (468 + 12)x - (936 + 60)y + (1170 - 36) = 0 \]
Thus,
\[ 233x^2 + 209y^2 - 456x - 996y + 1134 = 0 \]

### Final Equation
The equation of the ellipse is:
\[ 233x^2 + 209y^2 - 456x - 996y + 1134 = 0 \]

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