The line 4Y =X +8 cuts the curve XY = 4+ 2X at the points A and B. Find the exact length of ab.

Let's find the exact length of the line segment \( AB \) where the line \( 4y = x + 8 \) intersects the curve \( xy = 4 + 2x \).

### Step 1: Express the Line Equation in Standard Form
First, we'll rewrite the line equation \( 4y = x + 8 \) in terms of \( y \):

\[
y = \frac{x + 8}{4}
\]

### Step 2: Find the Points of Intersection
Substitute \( y = \frac{x + 8}{4} \) into the curve equation \( xy = 4 + 2x \):

\[
x\left(\frac{x + 8}{4}\right) = 4 + 2x
\]

Simplify this equation:

\[
\frac{x(x + 8)}{4} = 4 + 2x
\]

Multiply both sides by 4 to eliminate the fraction:

\[
x(x + 8) = 16 + 8x
\]

Expand and rearrange the equation:

\[
x^2 + 8x = 16 + 8x
\]

Subtract \( 8x \) from both sides:

\[
x^2 = 16
\]

Take the square root of both sides:

\[
x = \pm 4
\]

So, the \( x \)-coordinates of the points of intersection \( A \) and \( B \) are \( x_1 = 4 \) and \( x_2 = -4 \).

### Step 3: Find the Corresponding \( y \)-Coordinates
Using \( y = \frac{x + 8}{4} \), the corresponding \( y \)-coordinates are:

For \( x_1 = 4 \):

\[
y_1 = \frac{4 + 8}{4} = \frac{12}{4} = 3
\]

For \( x_2 = -4 \):

\[
y_2 = \frac{-4 + 8}{4} = \frac{4}{4} = 1
\]

### 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:

\[
AB = \sqrt{(-4 - 4)^2 + (1 - 3)^2}
\]
\[
AB = \sqrt{(-8)^2 + (-2)^2}
\]
\[
AB = \sqrt{64 + 4}
\]
\[
AB = \sqrt{68}
\]
\[
AB = 2\sqrt{17}
\]

### Final Answer
The exact length of \( AB \) is \( 2\sqrt{17} \).

---

Would you like more details on any part of this solution? Here are some related questions you might find helpful:

1. How do you simplify the square root of non-perfect squares like \( \sqrt{68} \)?
2. What would happen if the line had a different slope?
3. How does the quadratic nature of the curve affect the number of intersection points?
4. Can we use other methods to find the intersection points?
5. How would the process differ if the line equation was more complex?

**Tip:** When dealing with distances between points on a plane, always double-check the distance formula application to ensure accuracy.

Learn how to find the exact length of line segment AB where the line 4y = x + 8 intersects the curve xy = 4 + 2x. Detailed step-by-step solution with the application of the distance formula and quadratic equation solving. Suitable for high school students studying geometry and quadratic equations.

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