In coordinate space, there is a circle $C$ formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 6$ and the plane $x + 2 z - 5 = 0$. Let P be the point on circle $C$ with the minimum $y$-coordinate, and let Q be the foot of the perpendicular from point P to the $xy$-plane. For a point X moving on circle $C$, the maximum value of $| \overrightarrow { \mathrm { PX } } + \overrightarrow { \mathrm { QX } } | ^ { 2 }$ is $a + b \sqrt { 30 }$. Find the value of $10 ( a + b )$. (Here, $a$ and $b$ are rational numbers.) [4 points]
In coordinate space, there is a circle $C$ formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 6$ and the plane $x + 2 z - 5 = 0$. Let P be the point on circle $C$ with the minimum $y$-coordinate, and let Q be the foot of the perpendicular from point P to the $xy$-plane. For a point X moving on circle $C$, the maximum value of $| \overrightarrow { \mathrm { PX } } + \overrightarrow { \mathrm { QX } } | ^ { 2 }$ is $a + b \sqrt { 30 }$.
Find the value of $10 ( a + b )$. (Here, $a$ and $b$ are rational numbers.) [4 points]