the line 2x+5y=1 meet the curve x^2+5xy-4y^2+10=0 at point A and B find the coordintes of the midpoints of AB

To find the coordinates of the midpoint of the line segment joining points \( A \) and \( B \), where the line \( 2x + 5y = 1 \) meets the curve \( x^2 + 5xy - 4y^2 + 10 = 0 \), we follow these steps:

### Step 1: Parametrize the points of intersection
We substitute the equation of the line \( 2x + 5y = 1 \) into the equation of the curve to find the \( x \)- and \( y \)-coordinates of the points of intersection \( A \) and \( B \).

From the line equation \( 2x + 5y = 1 \), express \( x \) in terms of \( y \):
\[
x = \frac{1 - 5y}{2}
\]

### Step 2: Substitute into the curve equation
Substitute \( x = \frac{1 - 5y}{2} \) into the equation of the curve \( x^2 + 5xy - 4y^2 + 10 = 0 \):

\[
\left( \frac{1 - 5y}{2} \right)^2 + 5 \left( \frac{1 - 5y}{2} \right)y - 4y^2 + 10 = 0
\]

### Step 3: Solve for \( y \)
Expand and simplify the equation to find the values of \( y \) at the points of intersection, then back-substitute into \( x = \frac{1 - 5y}{2} \) to find the corresponding \( x \)-coordinates.

### Step 4: Find the midpoint
Once we have the coordinates of points \( A \) and \( B \), the midpoint is given by:
\[
M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right)
\]

Let me calculate the exact values for you.The coordinates of points \( A \) and \( B \) where the line \( 2x + 5y = 1 \) meets the curve \( x^2 + 5xy - 4y^2 + 10 = 0 \) are:
- \( A = (3, -1) \)
- \( B = (-2, 1) \)

The midpoint of \( AB \) is:
\[
M = \left( \frac{1}{2}, 0 \right)
\]

Would you like further details or clarification?

Here are five related questions to deepen your understanding:
1. How would the midpoint change if the curve equation was altered slightly?
2. Can you find the length of segment \( AB \)?
3. What would happen if the line equation was rotated or translated?
4. How can you determine if \( A \) and \( B \) are symmetrically positioned relative to an axis?
5. How would the solution differ if the curve were a circle?

**Tip**: The midpoint formula can also be used in three-dimensional geometry to find midpoints of lines in space.

The line 2x + 5y = 1 meets the curve x^2 + 5xy - 4y^2 + 10 = 0 at points A and B. Find the coordinates of the midpoint of AB.

Explore how to find the midpoint of the line segment formed by the intersection points of the line 2x + 5y = 1 and the curve x^2 + 5xy - 4y^2 + 10 = 0. This problem demonstrates key concepts in analytical geometry, including the system of equations and midpoint calculation. Ideal for Grades 10-12 students.

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