Hyperbola

$S=0$ represents a circle with radius $\sqrt{2}$, when $k=9$

$S=$ Orepresents an ellipse with eccentricity $\sqrt{\frac{2}{3}}$, when $k=10$

$S=0$ represents a hyperbola with eccentricity $\sqrt{\frac{6}{5}}$, when $k=12$

$S=0$ represents a hyperbola with eccentricity $\sqrt{\frac{3}{2}}$, when $k=13$