it is possible for two stars to have different temperatures and different sizes but the same luminosity. explain how.

Two stars (a and b) can have the same luminosity, but different surface area and temperature if the following condition is met:

(T_a^4) (R_a^2) = (T_b^4) (R_b^2)

Explanation:

The luminosity of a star is the total energy that produces in one second. It depends on the size of the star and its surface temperature.

L = σ (T^4) (4πR^2)

L is the luminosity f the star, T is the temperature of the surface of the star and R is its radius.

Two stars can have the same luminosity if the relation between the radius and the surface temperature is maintained.

To see this lets suposed you have 2 stars, a and b, and the luminosities of each one of them:

L_a = σ (T_a^4) (4πR_a^2)

L_b = σ (T_b^4) (4πR_b^2)

you can assume that L_a and L_b are equal:

σ (T_a^4) (4πR_a^2) = σ (T_b^4) (4πR_b^2)

Now, you can cancel the constants:

(T_a^4) (R_a^2) = (T_b^4) (R_b^2)

as long as this relation between a and b is true, then the luminosity can be the same.