A curve has the equation [tex]y = 2xy + 5[/tex] and a line has the equation [tex]2x + 5y = 1[/tex].

The curve and the line intersect at points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]. Find the coordinates of the midpoint of the line segment [tex]\(AB\)[/tex].



Answer :

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]