As shown in the figure, there is a line $l : x - y - 1 = 0$ and a hyperbola $C : x ^ { 2 } - 2 y ^ { 2 } = 1$ with one focus at point $\mathrm { F } ( c , 0 )$ (where $c < 0$). Under a rotation transformation by angle $\theta$ about the origin, the line $l$ is mapped to a line passing through the focus F of the hyperbola $C$. What is the value of $\sin 2 \theta$? [4 points] (1) $- \frac { 2 } { 3 }$ (2) $- \frac { 5 } { 9 }$ (3) $- \frac { 4 } { 9 }$ (4) $- \frac { 1 } { 3 }$ (5) $- \frac { 2 } { 9 }$
As shown in the figure, there is a line $l : x - y - 1 = 0$ and a hyperbola $C : x ^ { 2 } - 2 y ^ { 2 } = 1$ with one focus at point $\mathrm { F } ( c , 0 )$ (where $c < 0$).
Under a rotation transformation by angle $\theta$ about the origin, the line $l$ is mapped to a line passing through the focus F of the hyperbola $C$. What is the value of $\sin 2 \theta$? [4 points]\\
(1) $- \frac { 2 } { 3 }$\\
(2) $- \frac { 5 } { 9 }$\\
(3) $- \frac { 4 } { 9 }$\\
(4) $- \frac { 1 } { 3 }$\\
(5) $- \frac { 2 } { 9 }$