1. Given adjacent vertices of the rectangle

The two adjacent vertices are $A(-8,5), \quad B(6,5).$

Since they are adjacent and have the same $y$-coordinate, side $AB$ is horizontal.

Its length is $AB = |6-(-8)| = 14.$

2. Find the center of the rectangle

A rectangle inscribed in a circle has its diagonals passing through the center of the circle. Hence the center of the circle is also the center of the rectangle.

The diameter lies along the line $3y = x+7.$

So the center of the rectangle must lie on this line.

Also, since $AB$ is a side of the rectangle and is horizontal, the opposite side will also be horizontal. Therefore the center lies on the perpendicular bisector of $AB$.

Midpoint of $AB$ is $M\left(\frac{-8+6}{2},\frac{5+5}{2}\right)=(-1,5).$

Since $AB$ is horizontal, its perpendicular bisector is the vertical line $x=-1.$

Thus the center $O$ is the intersection of $x=-1$ and $3y=x+7.$

Substitute $x=-1$: $3y=-1+7=6 \implies y=2.$

Hence, $O=(-1,2).$

3. Find the other side length of the rectangle

The center of the rectangle is the midpoint between the top and bottom sides.

Since $AB$ lies on $y=5$, and the center has $y=2$, the vertical distance from the center to side $AB$ is $5-2=3.$

Therefore the full vertical side length of the rectangle is $2\times 3=6.$

4. Area of the rectangle

Thus the rectangle has dimensions $14 \times 6.$

So its area is $14\cdot 6 = 84.$

5. Check with options

The correct option is $\boxed{84}$ which is Option B.

6. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

So they agree.