In a homogeneous, isotropic elastic material, the modulus of elasticity (E) can be expressed in terms of two other elastic constants, shear modulus (G) and bulk modulus (K). The relationship between these constants can be derived from the laws of elasticity and the definition of modulus of elasticity.

Modulus of Elasticity (E):
- Modulus of elasticity (E) represents the stiffness or rigidity of a material.
- It measures the material's ability to resist deformation under an applied load.
- It is defined as the ratio of stress (σ) to strain (ε) in a material subjected to an external force.
- Mathematically, E = σ/ε.

Shear Modulus (G):
- Shear modulus (G) is a measure of the material's resistance to shear deformation.
- It represents the ratio of shear stress (τ) to shear strain (γ) in a material.
- Mathematically, G = τ/γ.

Bulk Modulus (K):
- Bulk modulus (K) is a measure of the material's resistance to uniform compression.
- It represents the ratio of normal stress (σ) to volumetric strain (εv) in a material.
- Mathematically, K = -σ/εv.

Deriving the Relationship:
To derive the relationship between E, G, and K, we need to consider the definitions of these elastic constants and the equations relating stress and strain.

1. Shear Modulus (G):
- From the definition of shear modulus, G = τ/γ.
- Rearranging, τ = Gγ.

2. Modulus of Elasticity (E):
- From the definition of modulus of elasticity, E = σ/ε.
- Rearranging, σ = Eε.

3. Bulk Modulus (K):
- From the definition of bulk modulus, K = -σ/εv.
- Rearranging, σ = -Kεv.

Substituting the values of σ from equations (2) and (3) into equation (1), we get:
Gγ = -Kεv.

Simplifying the equation:
G/εv = -K/γ.

Since εv = 3ε (for isotropic materials), and γ = √3ε (for simple shear),
G/3ε = -K/√3ε.

Cancelling out ε, we get:
G/3 = -K/√3.

Rearranging the equation, we have:
G = -3K/√3.

Simplifying further:
G = -3K√3/3.

Finally, multiplying both sides by -1 and simplifying, we obtain:
G = √3K.

Now, substituting the value of G in terms of K into the equation for E (E = σ/ε), we get:
E = σ/ε = (√3K)ε/ε = √3K.

Therefore, the modulus of elasticity (E) in terms of G and K is equal to √3K. So, option 'C' is the correct answer.