To determine the coordinates of the midpoint of the line segment connecting points [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to follow these steps:
### Step 1: Equation Setup
We are given the equations for the curve and the line:
1. [tex]\( y = 2xy + 5 \)[/tex]
2. [tex]\( 2x + 5y = 1 \)[/tex]
### Step 2: Solving the System of Equations
We must solve these two equations simultaneously to find the points of intersection [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
### Step 3: Points of Intersection
After solving the system of equations, we find the points of intersection are:
[tex]\[ A = (-2, 1) \][/tex]
[tex]\[ B = (3, -1) \][/tex]
### Step 4: Midpoint Calculation
The midpoint [tex]\( M \)[/tex] of a line segment connecting two points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] is calculated using the midpoint formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Substitute the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] into the formula:
[tex]\[ M = \left( \frac{-2 + 3}{2}, \frac{1 + (-1)}{2} \right) \][/tex]
### Step 5: Simplifying the Midpoint Coordinates
Simplify the expressions inside the midpoint formula:
[tex]\[ M = \left( \frac{1}{2}, \frac{0}{2} \right) \][/tex]
[tex]\[ M = \left( \frac{1}{2}, 0 \right) \][/tex]
### Conclusion
The coordinates of the midpoint of the line segment [tex]\( AB \)[/tex] are:
[tex]\[ M = \left( \frac{1}{2}, 0 \right) \][/tex]
