The line for why equals X +8 cuts the curve XY equals 4+ 2X at the points A and B find the exact length of a B

To find the exact length of the line segment \( AB \) where the line \( y = x + 8 \) intersects the curve \( xy = 4 + 2x \), we will proceed with the following steps:

### Step 1: Find the Points of Intersection
The points of intersection occur where both equations are satisfied simultaneously. Substituting \( y = x + 8 \) into the curve equation \( xy = 4 + 2x \):

\[
x(x + 8) = 4 + 2x
\]

This simplifies to:

\[
x^2 + 8x = 4 + 2x
\]

Move all terms to one side of the equation:

\[
x^2 + 6x - 4 = 0
\]

### Step 2: Solve the Quadratic Equation
We solve the quadratic equation \( x^2 + 6x - 4 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 6 \), and \( c = -4 \):

\[
x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-4)}}{2(1)}
\]
\[
x = \frac{-6 \pm \sqrt{36 + 16}}{2}
\]
\[
x = \frac{-6 \pm \sqrt{52}}{2}
\]
\[
x = \frac{-6 \pm 2\sqrt{13}}{2}
\]
\[
x = -3 \pm \sqrt{13}
\]

So, the \( x \)-coordinates of the points of intersection \( A \) and \( B \) are \( x_1 = -3 + \sqrt{13} \) and \( x_2 = -3 - \sqrt{13} \).

### Step 3: Find the Corresponding \( y \)-Coordinates
Using \( y = x + 8 \), the corresponding \( y \)-coordinates are:

For \( x_1 = -3 + \sqrt{13} \):

\[
y_1 = (-3 + \sqrt{13}) + 8 = 5 + \sqrt{13}
\]

For \( x_2 = -3 - \sqrt{13} \):

\[
y_2 = (-3 - \sqrt{13}) + 8 = 5 - \sqrt{13}
\]

### Step 4: Calculate the Distance \( AB \)
The length of the segment \( AB \) is given by the distance formula:

\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the values:

\[
AB = \sqrt{\left[(-3 - \sqrt{13}) - (-3 + \sqrt{13})\right]^2 + \left[(5 - \sqrt{13}) - (5 + \sqrt{13})\right]^2}
\]
\[
AB = \sqrt{\left[-2\sqrt{13}\right]^2 + \left[-2\sqrt{13}\right]^2}
\]
\[
AB = \sqrt{4 \times 13 + 4 \times 13}
\]
\[
AB = \sqrt{52 + 52}
\]
\[
AB = \sqrt{104}
\]
\[
AB = 2\sqrt{26}
\]

### Final Answer
The exact length of \( AB \) is \( 2\sqrt{26} \).

---

Would you like to go over any specific part of this solution in more detail? Here are some related questions you might find helpful:

1. How do you derive the quadratic formula?
2. Can you explain how to apply the distance formula in more detail?
3. What is the geometric interpretation of the solution?
4. How would the solution change if the line equation was different?
5. What are other methods to solve this problem?

**Tip:** When solving for the points of intersection between a line and a curve, substituting the line equation into the curve's equation is a common strategy.

Learn how to calculate the exact length of segment AB where the line y = x + 8 intersects the curve xy = 4 + 2x. This detailed solution covers quadratic equations, intersection of a line and curve, and the distance formula. Suitable for advanced high school students (Grades 10-12).

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