Given, the two adjacent vertices of the rectangle are $\left(-8, 5\right)$ and $\left(6, 5\right),$ the $y$-co-ordinates of both of these points is same this means they lie on the line $y=5,$ which is parallel to the $x$-axis.

And, we know that the sides of a rectangle are perpendicular to each other, hence the other two sides passing through these vertices are parallel to $y$-axis.

Also, we know that the equation of a line parallel $y$-axis is $x=k.$

Therefore, the equation of the sides of the rectangle passing through $\left(-8, 5\right)$ and $\left(6, 5\right)$ are respectively $x=-8$ and $x=6.$

And, hence the other two vertices of the rectangle can be taken as $\left(6, a\right)$ and $\left(-8, b\right).$

And since the other side is parallel to $x$-axis, hence $a=b.$

Here, diagonal subtends right angle at the circumference of circle, so its mid-point is the same as centre of circle, which lies on the given diameter.

The mid-point of a line segment joining the points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right),$ hence, the mid-point of the diagonal of the rectangle is $\left(\frac{-8+6}{2}, \frac{a+5}{2}\right)=\left(-1, \frac{a+5}{2}\right)$

This point lie on the diameter $3y=x+7$

So, $\frac{3\left(a+5\right)}{2}=-1+7$
$\Rightarrow a+5=\frac{2\left(6\right)}{3}$

$\Rightarrow a=-1$

Hence, the vertices are $\left(-8, 5\right), \left(6, 5\right), \left(6, -1\right)$ and $\left(-8, -1\right).$

The distance between the points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ is $\sqrt{{\left(x_1-x_2\right)}^2+{\left(y_1-y_2\right)}^2}$

Thus, the length of sides are $6$ and $14$ units.

Hence, area $=l\times b=14\times 6=84$ sq. units