As shown in the figure, in the coordinate plane, for two points $\mathrm { A } , \mathrm { B }$ on the $x$-axis, the parabola $p _ { 1 }$ with vertex at A and the parabola $p _ { 2 }$ with vertex at B satisfy the following conditions. What is the area of triangle ABC? [4 points] (가) The focus of $p _ { 1 }$ is B, and the focus of $p _ { 2 }$ is the origin O. (나) $p _ { 1 }$ and $p _ { 2 }$ meet at two points $\mathrm { C } , \mathrm { D }$ on the $y$-axis. (다) $\overline { \mathrm { AB } } = 2$ (1) $4 ( \sqrt { 2 } - 1 )$ (2) $3 ( \sqrt { 3 } - 1 )$ (3) $2 ( \sqrt { 5 } - 1 )$ (4) $\sqrt { 3 } + 1$ (5) $\sqrt { 5 } + 1$
As shown in the figure, in the coordinate plane, for two points $\mathrm { A } , \mathrm { B }$ on the $x$-axis, the parabola $p _ { 1 }$ with vertex at A and the parabola $p _ { 2 }$ with vertex at B satisfy the following conditions. What is the area of triangle ABC? [4 points]\\
(가) The focus of $p _ { 1 }$ is B, and the focus of $p _ { 2 }$ is the origin O.\\
(나) $p _ { 1 }$ and $p _ { 2 }$ meet at two points $\mathrm { C } , \mathrm { D }$ on the $y$-axis.\\
(다) $\overline { \mathrm { AB } } = 2$\\
(1) $4 ( \sqrt { 2 } - 1 )$\\
(2) $3 ( \sqrt { 3 } - 1 )$\\
(3) $2 ( \sqrt { 5 } - 1 )$\\
(4) $\sqrt { 3 } + 1$\\
(5) $\sqrt { 5 } + 1$