Find the equation to the parabola whose focus is `(-2,1)` and directrix is `6x-3y=8`.

Answer

To find the equation of the parabola with focus at \((-2, 1)\) and directrix given by the line \(6x - 3y = 8\), 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 directrix is given by the equation \(6x - 3y - 8 = 0\).

### Step 2: Write the distance formula
According to the definition of a parabola, the distance from any point \(P(x, y)\) on the parabola to the focus \(F\) is equal to the distance from \(P\) to the directrix. Therefore, we can express this as:
\[ PF = PM \]
where \(PF\) is the distance from point \(P\) to the focus and \(PM\) is the distance from point \(P\) to the directrix.

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

### Step 4: Calculate the distance from \(P\) to the directrix
The distance \(PM\) from point \(P(x, y)\) to the directrix \(6x - 3y - 8 = 0\) can be calculated using the formula:
\[ PM = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where \(A = 6\), \(B = -3\), and \(C = -8\). Thus,
\[ PM = \frac{|6x - 3y - 8|}{\sqrt{6^2 + (-3)^2}} = \frac{|6x - 3y - 8|}{\sqrt{36 + 9}} = \frac{|6x - 3y - 8|}{\sqrt{45}} = \frac{|6x - 3y - 8|}{3\sqrt{5}} \]

### Step 5: Set the distances equal
Now, we set the two distances equal:
\[ \sqrt{(x + 2)^2 + (y - 1)^2} = \frac{|6x - 3y - 8|}{3\sqrt{5}} \]

### Step 6: Square both sides
Squaring both sides to eliminate the square root gives:
\[ (x + 2)^2 + (y - 1)^2 = \left(\frac{6x - 3y - 8}{3\sqrt{5}}\right)^2 \]
This simplifies to:
\[ (x + 2)^2 + (y - 1)^2 = \frac{(6x - 3y - 8)^2}{45} \]

### Step 7: Multiply through by 45
To eliminate the fraction, multiply both sides by 45:
\[ 45((x + 2)^2 + (y - 1)^2) = (6x - 3y - 8)^2 \]

### Step 8: Expand both sides
Expanding the left side:
\[ 45((x^2 + 4x + 4) + (y^2 - 2y + 1)) = 45x^2 + 180x + 180 + 45y^2 - 90y \]
Expanding the right side:
\[ (6x - 3y - 8)^2 = 36x^2 - 36xy - 96x + 9y^2 + 48y + 64 \]

### Step 9: Rearrange the equation
Now, rearranging gives us:
\[ 45x^2 + 45y^2 + 180x - 90y + 225 = 36x^2 + 9y^2 - 36xy - 96x + 48y + 64 \]

### Step 10: Combine like terms
Combining like terms results in:
\[ (45x^2 - 36x^2) + (45y^2 - 9y^2) + (180x + 96x) + (-90y - 48y) + (225 - 64) = 0 \]
This simplifies to:
\[ 9x^2 + 36y^2 + 276x - 138y + 161 = 0 \]

### Final Equation
Thus, the equation of the parabola is:
\[ 9x^2 + 36y^2 + 276x - 138y + 161 = 0 \]