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 a focus at \((-1, -2)\) and a directrix given by the line \(x - 2y + 3 = 0\), we can follow these steps:

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

### Step 2: Rewrite the directrix in slope-intercept form
To make calculations easier, we can rewrite the directrix in slope-intercept form (y = mx + b):
\[ x - 2y + 3 = 0 \implies 2y = x + 3 \implies y = \frac{1}{2}x + \frac{3}{2} \]
This gives us the slope \(m = \frac{1}{2}\) and the y-intercept \(\frac{3}{2}\).

### Step 3: Use the definition of a parabola
The definition of a parabola states that for any point \((x, y)\) on the parabola, the distance to the focus is equal to the distance to the directrix.

The distance from a point \((x, y)\) to the focus \((-1, -2)\) is:
\[ d_{focus} = \sqrt{(x + 1)^2 + (y + 2)^2} \]

The distance from the point \((x, y)\) to the directrix \(x - 2y + 3 = 0\) is given by the formula:
\[ d_{directrix} = \frac{|x - 2y + 3|}{\sqrt{1^2 + (-2)^2}} = \frac{|x - 2y + 3|}{\sqrt{5}} \]

### Step 4: Set the distances equal
According to the definition of a parabola, we set these distances equal:
\[ \sqrt{(x + 1)^2 + (y + 2)^2} = \frac{|x - 2y + 3|}{\sqrt{5}} \]

### Step 5: 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 6: Multiply through by 5 to eliminate the fraction
\[ 5[(x + 1)^2 + (y + 2)^2] = (x - 2y + 3)^2 \]

### Step 7: Expand both sides
Expanding the left side:
\[ 5[(x^2 + 2x + 1) + (y^2 + 4y + 4)] = 5x^2 + 10x + 5 + 5y^2 + 20y \]
Expanding the right side:
\[ (x - 2y + 3)^2 = x^2 - 4xy + 4y^2 + 6x - 12y + 9 \]

### Step 8: Set the equation to zero
Now we have:
\[ 5x^2 + 5y^2 + 10x + 20y + 5 = x^2 - 4xy + 4y^2 + 6x - 12y + 9 \]
Rearranging gives:
\[ 4x^2 + y^2 + 4xy + 4x + 32y - 4 = 0 \]

### Step 9: Final equation
This is the equation of the parabola.