Typical Geometry

Let the diagonals of rhombus be $BD = 2a, AC = 2b$ with $a \leq b$ . Since the longer diagonal is greater than the side of rhombus. It means radius of larger circle is greater than distance between the centers of both circles.So two circles can only be tangent to each other internally. The condition for internal tangency of two circles is:

Difference of radius of both circles = distance between their centers.

$\Rightarrow AC - BD = 10$

$\Rightarrow 2b - 2a = 10$

$\Rightarrow b - a = 5$

Also,

$a^2 + b^2 = 10^2 = 100$

$\Rightarrow (b - a)^2 + 2ab = 100$

$\Rightarrow 5^2 + 2ab = 100$

$\Rightarrow 2ab = 100 - 25 = 75$

Area of rhombus $=$ Half of product of diagonals.

$\Rightarrow$ Area $= \Large\frac{1}{2}$ $(2a)(2b)$ $= 2ab = 75$