K-Points on Conics

Say an ellipse or hyperbola has a $K$-point if there is a point $P$ on it whose distances to the two foci are in ratio $2 : 1$. Determine, with proof, which of the following curves have a $K$-point:

1. $\dfrac{x^2}{16} + \dfrac{y^2}{15} = 1$; 2. $\dfrac{x^2}{25} + \dfrac{y^2}{24} = 1$; 3. $x^2 - \dfrac{y^2}{15} = 1$; 4. $x^2 - y^2 = 1$.