JEE PHYSICS MCQ: MOTION IN PLANE

Consider three vectors $\mathbf{A}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{B}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{C}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$. A vector $\mathbf{X}$ of the form $\alpha \mathbf{A}+\beta \mathbf{B}$ ( $\alpha$ and $\beta$ are numbers) is perpendicular to $\mathbf{C}$. The ratio of $\alpha$ and $\beta$ is