The equation of the diirectrix of the parabola is `3x+2y+1-0`. The focus is `(2,1)`. Find the equation of the parabola.

Answer

To find the equation of the parabola given the directrix \(3x + 2y + 1 = 0\) and the focus \((2, 1)\), we can follow these steps:

### Step 1: Identify the focus and directrix
The focus of the parabola is given as \(F(2, 1)\) and the equation of the directrix is \(3x + 2y + 1 = 0\).

### Step 2: Set up the distance formula
Let \(P(x, y)\) be a point on the parabola. The distance from point \(P\) to the focus \(F(2, 1)\) is given by:
\[ d_F = \sqrt{(x - 2)^2 + (y - 1)^2} \]

### Step 3: Find the distance from point \(P\) to the directrix
The distance from point \(P(x, y)\) to the directrix \(3x + 2y + 1 = 0\) can be calculated using the formula for the distance from a point to a line:
\[ d_D = \frac{|3x + 2y + 1|}{\sqrt{3^2 + 2^2}} = \frac{|3x + 2y + 1|}{\sqrt{13}} \]

### Step 4: Set the distances equal
For a parabola, the distance from the point \(P\) to the focus equals the distance from \(P\) to the directrix. Therefore, we have:
\[ \sqrt{(x - 2)^2 + (y - 1)^2} = \frac{|3x + 2y + 1|}{\sqrt{13}} \]

### Step 5: Square both sides
To eliminate the square root, we square both sides:
\[ (x - 2)^2 + (y - 1)^2 = \frac{(3x + 2y + 1)^2}{13} \]

### Step 6: Multiply through by 13
Multiplying both sides by 13 gives:
\[ 13((x - 2)^2 + (y - 1)^2) = (3x + 2y + 1)^2 \]

### Step 7: Expand both sides
Expanding the left side:
\[ 13((x - 2)^2 + (y - 1)^2) = 13((x^2 - 4x + 4) + (y^2 - 2y + 1)) = 13x^2 - 52x + 52 + 13y^2 - 26y + 13 \]
This simplifies to:
\[ 13x^2 + 13y^2 - 52x - 26y + 65 \]

Expanding the right side:
\[ (3x + 2y + 1)^2 = 9x^2 + 4y^2 + 1 + 12xy + 6x + 4y \]

### Step 8: Set the equation to zero
Now we set the equation:
\[ 13x^2 + 13y^2 - 52x - 26y + 65 = 9x^2 + 4y^2 + 1 + 12xy + 6x + 4y \]
Rearranging gives:
\[ (13x^2 - 9x^2) + (13y^2 - 4y^2) - (52x - 6x) - (26y - 4y) + (65 - 1) - 12xy = 0 \]
This simplifies to:
\[ 4x^2 + 9y^2 - 12xy - 58x - 30y + 64 = 0 \]

### Final Equation of the Parabola
Thus, the equation of the parabola is:
\[ 4x^2 + 9y^2 - 12xy - 58x - 30y + 64 = 0 \]

---