On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin. It is known that a point $P$ on $\Gamma$ is transformed by this rotation to a point $P ^ { \prime }$ that falls on the $x$-axis, and the $x$-coordinate of $P ^ { \prime }$ is positive. Find the coordinates of point $P$.
On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
It is known that a point $P$ on $\Gamma$ is transformed by this rotation to a point $P ^ { \prime }$ that falls on the $x$-axis, and the $x$-coordinate of $P ^ { \prime }$ is positive. Find the coordinates of point $P$.