Let’s solve the given problem step-by-step.
Problem Overview
We aim to find the value of , where:
- is the area of the rectangle.
- is the maximum perimeter of the rectangle formed by vertices , , , and .
The points and lie on the curves:
- (Quadrant II).
- (Quadrant I).
The points and lie on the x-axis.
Step 1: Coordinates of the Points
- for .
- for .
Step 2: Dimensions of the Rectangle
The height of the rectangle is determined by the -coordinates of and :
The width of the rectangle is determined by the distance between and , which is:
Step 3: Area of the Rectangle
The area of the rectangle is given by:
Step 4: Perimeter of the Rectangle
The perimeter of the rectangle is:
Step 5: Maximize the Perimeter
We maximize by choosing values for and such that the rectangle covers the largest possible area under the curves. In this scenario:
- The maximum width is achieved when and .
Thus:
The corresponding height is:
Step 6: Compute and
Using the maximum dimensions:
- Area:
- Perimeter:
Step 7: Compute
Final Answer: