To find the equation of the parabola with focus at (-1, -2) and 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 \(F(-1, -2)\). The equation of the directrix is given as \(x - 2y + 3 = 0\).
### Step 2: Rewrite the directrix in slope-intercept form
We can rewrite the directrix equation in the form \(y = mx + b\):
\[
x - 2y + 3 = 0 \implies 2y = x + 3 \implies y = \frac{1}{2}x + \frac{3}{2}
\]
This shows that the slope \(m\) is \(\frac{1}{2}\) and the y-intercept is \(\frac{3}{2}\).
### Step 3: Determine the vertex of the parabola
The vertex of the parabola lies halfway between the focus and the directrix. To find the vertex, we need to calculate the perpendicular distance from the focus to the directrix.
### Step 4: Find the distance from the focus to the directrix
The distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by:
\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
For our directrix \(x - 2y + 3 = 0\) (where \(A = 1\), \(B = -2\), \(C = 3\)) and the focus at \((-1, -2)\):
\[
d = \frac{|1(-1) - 2(-2) + 3|}{\sqrt{1^2 + (-2)^2}} = \frac{|-1 + 4 + 3|}{\sqrt{1 + 4}} = \frac{|6|}{\sqrt{5}} = \frac{6}{\sqrt{5}}
\]
### Step 5: Find the coordinates of the vertex
The vertex \(V\) is located at a distance of \(\frac{d}{2}\) from the focus towards the directrix. The direction vector from the focus to the line can be found using the slope of the directrix. The slope of the line is \(\frac{1}{2}\), so the direction vector can be represented as \((2, 1)\).
To find the unit vector in this direction:
\[
\text{Magnitude} = \sqrt{2^2 + 1^2} = \sqrt{5}
\]
The unit vector is:
\[
\left(\frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}}\right)
\]
Now, moving from the focus \((-1, -2)\) towards the directrix by \(\frac{d}{2} = \frac{3}{\sqrt{5}}\):
\[
V = \left(-1 + \frac{3}{\sqrt{5}} \cdot \frac{2}{\sqrt{5}}, -2 + \frac{3}{\sqrt{5}} \cdot \frac{1}{\sqrt{5}}\right)
\]
Calculating the coordinates of the vertex gives:
\[
V = \left(-1 + \frac{6}{5}, -2 + \frac{3}{5}\right) = \left(-\frac{5}{5} + \frac{6}{5}, -\frac{10}{5} + \frac{3}{5}\right) = \left(\frac{1}{5}, -\frac{7}{5}\right)
\]
### Step 6: Use the standard form of the parabola
The standard form of a parabola with vertex \((h, k)\) and focus \((h, k + p)\) is:
\[
(x - h)^2 = 4p(y - k)
\]
Here, \(p\) is the distance from the vertex to the focus. Since the focus is below the vertex, \(p\) will be negative.
Calculating \(p\):
\[
p = -\frac{6}{\sqrt{5}} \div 2 = -\frac{3}{\sqrt{5}}
\]
### Step 7: Substitute into the parabola equation
Substituting \(h = \frac{1}{5}\), \(k = -\frac{7}{5}\), and \(p = -\frac{3}{\sqrt{5}}\):
\[
\left(x - \frac{1}{5}\right)^2 = 4\left(-\frac{3}{\sqrt{5}}\right)\left(y + \frac{7}{5}\right)
\]
Simplifying gives:
\[
\left(x - \frac{1}{5}\right)^2 = -\frac{12}{\sqrt{5}}(y + \frac{7}{5})
\]
### Final Equation
The equation of the parabola is:
\[
\left(x - \frac{1}{5}\right)^2 = -\frac{12}{\sqrt{5}}(y + \frac{7}{5})
\]
