To solve the problem, we need to analyze the given tangents and their relationship with the parabola. The tangents are given by the equations \( x + y = 2 \) and \( x - y = 2 \), and they touch the parabola at the points \( (1, 1) \) and \( (4, 2) \) respectively.
### Step 1: Find the point of intersection of the tangents
We start by solving the two equations to find their point of intersection.
1. The equations are:
\[
x + y = 2 \quad (1)
\]
\[
x - y = 2 \quad (2)
\]
2. We can add both equations (1) and (2):
\[
(x + y) + (x - y) = 2 + 2
\]
This simplifies to:
\[
2x = 4 \implies x = 2
\]
3. Now substitute \( x = 2 \) back into equation (1) to find \( y \):
\[
2 + y = 2 \implies y = 0
\]
So, the point of intersection is \( (2, 0) \).
### Step 2: Find the midpoint of the points where the tangents touch the parabola
Next, we find the midpoint of the points \( (1, 1) \) and \( (4, 2) \).
1. The midpoint \( M \) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
where \( (x_1, y_1) = (1, 1) \) and \( (x_2, y_2) = (4, 2) \).
2. Calculating the midpoint:
\[
M = \left( \frac{1 + 4}{2}, \frac{1 + 2}{2} \right) = \left( \frac{5}{2}, \frac{3}{2} \right)
\]
### Step 3: Find the slope of the axis of the parabola
Now we need to find the slope of the line connecting the point of intersection \( (2, 0) \) and the midpoint \( \left( \frac{5}{2}, \frac{3}{2} \right) \).
1. The slope \( m \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \( (x_1, y_1) = (2, 0) \) and \( (x_2, y_2) = \left( \frac{5}{2}, \frac{3}{2} \right) \).
2. Substituting the values:
\[
m = \frac{\frac{3}{2} - 0}{\frac{5}{2} - 2} = \frac{\frac{3}{2}}{\frac{1}{2}} = 3
\]
### Step 4: Find the equation of the directrix
Since the slope of the axis is \( 3 \), the slope of the directrix (which is perpendicular to the axis) will be the negative reciprocal, which is \( -\frac{1}{3} \).
1. The equation of the directrix can be written in point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Using the point \( (2, 0) \) and slope \( -\frac{1}{3} \):
\[
y - 0 = -\frac{1}{3}(x - 2)
\]
Simplifying gives:
\[
y = -\frac{1}{3}x + \frac{2}{3}
\]
Rearranging:
\[
x + 3y = 2
\]
### Step 5: Verify the focus and check the options
To find the focus, we need to find the distances from the points \( (1, 1) \) and \( (4, 2) \) to the directrix and use the section formula to find the focus.
1. The distances from points \( A(1, 1) \) and \( B(4, 2) \) to the directrix can be calculated using the formula for the distance from a point to a line.
2. After calculating the distances and applying the section formula, we find the coordinates of the focus.
3. Finally, we check which of the given options are correct based on the calculated focus and the equation of the directrix.
