Let $M=\{(t^5,t^3,t):t\in\mathbb{R}\}$, and suppose $\lambda$ has $ax+by+cz+d=0$ as it's cartesian equation (with $a,b,c$ not all zero). Then $M\cap \lambda$ is nonempty because $at^5+bt^3+ct+d=0$ always has a real solution (because a polynomial of odd degree always has a real root), and it's finite because $at^5+bt^3+ct+d=0$ has a finite number of solutions (because a non-constant polynomial has a finite number of roots).

Let $M=\{(t^5,t^3,t):t\in\mathbb{R}\}$, and suppose $\lambda$ has $ax+by+cz+d=0$ as it's cartesian equation (with $a,b,c$ not all zero). Then $M\cap \lambda$ is nonempty because $at^5+bt^3+ct+d=0$ always has a real solution (because a polynomial of odd degree always has a real root), and it's finite because $at^5+bt^3+ct+d=0$ has a finite number of solutions (because a non-constant polynomial has a finite number of roots).