Answer
Statement (b) is true.
Work Step by Step
By Theorem 1, there is a scalar $\lambda$ such that $\nabla {f_P} = \lambda \nabla {g_P}$.
This implies that $\nabla {f_P}$ is parallel to $\nabla {g_P}$.
Since $\nabla {g_P}$ is orthogonal to $g\left( {x,y} \right) = 0$ at $P$, it follows that $\nabla {f_P}$ is orthogonal to $g\left( {x,y} \right) = 0$ at $P$. So, statement (b) is true.
