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The image contains a mathematical problem, and the following is the content transcribed from the image:

\[
x^2 + y^2 = 64
\]
\[
y = 5
\]
The question asks:

**Q.** Determine the area covered by the circle and the straight line from the above information.

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Let's break this down:

1. The equation \(x^2 + y^2 = 64\) represents a circle with radius \(r = 8\), centered at the origin \((0,0)\).
2. The line \(y = 5\) is a horizontal line.

To find the area covered between the circle and the line, we can:

1. Find the points of intersection between the circle and the line.
2. Use integration to calculate the area of the segment formed between these points.

Let me calculate the area for you.The points of intersection between the circle and the line \( y = 5 \) are \( x = -\sqrt{39} \) and \( x = \sqrt{39} \).

The area of the segment formed between the circle and the line \( y = 5 \) is:

\[
\text{Area} = -5\sqrt{39} + 64 \cdot \arcsin\left(\frac{\sqrt{39}}{8}\right)
\]

This represents the exact area of the region enclosed by the circle and the line.

Would you like a detailed explanation of the steps or further assistance? Here are some related questions:

1. How can I compute the decimal value of this area?
2. What is the geometric significance of the arcsine term in the result?
3. Can this method be generalized for other lines intersecting a circle?
4. How do we handle cases where the line doesn't intersect the circle?
5. What are the properties of circle segments in other contexts?

**Tip:** For more complex shapes, using symmetry often simplifies area calculations!

Determine the area covered by the circle and the straight line from the information provided:
Equation of the circle: x^2 + y^2 = 64
Equation of the line: y = 5

Learn how to calculate the area enclosed between a circle with equation x^2 + y^2 = 64 and a horizontal line y = 5. This problem involves finding intersection points, using integration, and applying geometry theorems to find the exact area of the circle segment.

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