To find the equation of the tangent at the vertex of the parabola given its focus and directrix, we can follow these steps:
### Step 1: Identify the Focus and Directrix
The focus of the parabola is given as \( F(1, 2) \) and the directrix is given by the equation \( x + 2y + 9 = 0 \).
### Step 2: Find the Foot of the Perpendicular from the Focus to the Directrix
To find the foot of the perpendicular \( M \) from the focus \( F(1, 2) \) to the directrix \( x + 2y + 9 = 0 \), we can use the formula for the foot of the perpendicular from a point to a line.
The line can be expressed in the standard form \( Ax + By + C = 0 \) where \( A = 1, B = 2, C = 9 \).
Using the formula:
\[
\frac{h - x_1}{a} = \frac{k - y_1}{b} = \frac{-Ax_1 - By_1 - C}{A^2 + B^2}
\]
where \( (x_1, y_1) = (1, 2) \) and \( (h, k) \) are the coordinates of point \( M \).
Substituting the values:
\[
\frac{h - 1}{1} = \frac{k - 2}{2} = \frac{-1 \cdot 1 - 2 \cdot 2 - 9}{1^2 + 2^2}
\]
Calculating the right-hand side:
\[
= \frac{-1 - 4 - 9}{1 + 4} = \frac{-14}{5} = -\frac{14}{5}
\]
Now we can equate:
1. \( h - 1 = -\frac{14}{5} \) implies \( h = -\frac{14}{5} + 1 = -\frac{14}{5} + \frac{5}{5} = -\frac{9}{5} \)
2. \( k - 2 = -\frac{14}{10} \) implies \( k = -\frac{14}{10} + 2 = -\frac{14}{10} + \frac{20}{10} = \frac{6}{10} = -\frac{18}{5} \)
Thus, the coordinates of \( M \) are \( \left(-\frac{9}{5}, -\frac{18}{5}\right) \).
### Step 3: Find the Vertex of the Parabola
The vertex \( V \) of the parabola is the midpoint of the segment \( FM \):
\[
V = \left( \frac{x_F + x_M}{2}, \frac{y_F + y_M}{2} \right) = \left( \frac{1 - \frac{9}{5}}{2}, \frac{2 - \frac{18}{5}}{2} \right)
\]
Calculating:
\[
x_V = \frac{5 - 9}{10} = -\frac{4}{10} = -\frac{2}{5}
\]
\[
y_V = \frac{10 - 18}{10} = -\frac{8}{10} = -\frac{4}{5}
\]
Thus, the vertex \( V \) is \( \left(-\frac{2}{5}, -\frac{4}{5}\right) \).
### Step 4: Equation of the Tangent at the Vertex
The tangent at the vertex of a parabola is parallel to the directrix. Since the directrix is given by \( x + 2y + 9 = 0 \), the tangent will have the same coefficients for \( x \) and \( y \).
Thus, the equation of the tangent can be written as:
\[
x + 2y + c = 0
\]
To find \( c \), we substitute the vertex coordinates into the equation:
\[
-\frac{2}{5} + 2\left(-\frac{4}{5}\right) + c = 0
\]
This simplifies to:
\[
-\frac{2}{5} - \frac{8}{5} + c = 0 \implies c = \frac{10}{5} = 2
\]
Thus, the equation of the tangent is:
\[
x + 2y + 2 = 0
\]
### Final Answer
The equation of the tangent at the vertex is:
\[
\boxed{x + 2y + 2 = 0}
\]
