Find the equation of each of the following parabolas.
focus at `(-1,-2)`, directrix `x-2y+3=0`.

Answer

To find the equation of the parabola with focus at \((-1, -2)\) and directrix given by the line \(x - 2y + 3 = 0\), we will use the focal directrix property. This property states that for any point \(P\) on the parabola, the distance from \(P\) to the focus \(S\) is equal to the perpendicular distance from \(P\) to the directrix \(L\).

### Step 1: Identify the focus and directrix
- Focus \(S = (-1, -2)\)
- Directrix \(L: x - 2y + 3 = 0\)

### Step 2: Write the coordinates of point \(P\)
Let \(P\) be a point on the parabola with coordinates \((x, y)\).

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

### Step 4: Calculate the perpendicular distance \(PL\)
The perpendicular distance \(PL\) from point \(P\) to the line \(L\) can be calculated using the formula:
\[ PL = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
where \(Ax + By + C = 0\) is the equation of the line. Here, \(A = 1\), \(B = -2\), and \(C = 3\). Thus:
\[ PL = \frac{|1 \cdot x - 2 \cdot y + 3|}{\sqrt{1^2 + (-2)^2}} = \frac{|x - 2y + 3|}{\sqrt{5}} \]

### Step 5: Set the distances equal
According to the focal directrix property:
\[ PS = PL \]
Thus, we have:
\[ \sqrt{(x + 1)^2 + (y + 2)^2} = \frac{|x - 2y + 3|}{\sqrt{5}} \]

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

### Step 7: Multiply through by 5 to eliminate the fraction
\[ 5[(x + 1)^2 + (y + 2)^2] = (x - 2y + 3)^2 \]

### Step 8: Expand both sides
Expanding the left side:
\[ 5[(x^2 + 2x + 1) + (y^2 + 4y + 4)] = 5x^2 + 10x + 5 + 5y^2 + 20y + 20 = 5x^2 + 5y^2 + 10x + 20y + 25 \]

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

### Step 9: Set the expanded forms equal
Equating both sides:
\[ 5x^2 + 5y^2 + 10x + 20y + 25 = x^2 - 4xy + 4y^2 + 6x - 12y + 9 \]

### Step 10: Rearrange the equation
Rearranging gives:
\[ 5x^2 - x^2 + 5y^2 - 4y^2 + 10x - 6x + 20y + 12y + 25 - 9 + 4xy = 0 \]
This simplifies to:
\[ 4x^2 + y^2 + 4xy + 4x + 32y + 16 = 0 \]

### Final Equation
Thus, the equation of the parabola is:
\[ 4x^2 + y^2 + 4xy + 4x + 32y + 16 = 0 \]