Two lines $\ell _ { 1 }$ and $\ell _ { 2 }$ in 3-dimensional space are given by $$\ell _ { 1 } = \{ ( t - 9 , - t + 7 , 6 ) \mid t \in \mathbb { R } \} \quad \text{and} \quad \ell _ { 2 } = \{ ( 7 , s + 3 , 3 s + 4 ) \mid s \in \mathbb { R } \}.$$ Questions (31) The plane passing through the origin and not intersecting either of $\ell _ { 1 }$ and $\ell _ { 2 }$ has equation $ax + by + cz = d$. Write the value of $| a + b + c + d |$ where $a, b, c, d$ are integers with $\gcd = 1$. (32) Let $r$ be the smallest possible radius of a circle that has a point on $\ell _ { 1 }$ as well as a point on $\ell _ { 2 }$. It is given that $r ^ { 2 }$ (i.e., the square of the smallest radius) is an integer. Write the value of $r ^ { 2 }$.
Two lines $\ell _ { 1 }$ and $\ell _ { 2 }$ in 3-dimensional space are given by
$$\ell _ { 1 } = \{ ( t - 9 , - t + 7 , 6 ) \mid t \in \mathbb { R } \} \quad \text{and} \quad \ell _ { 2 } = \{ ( 7 , s + 3 , 3 s + 4 ) \mid s \in \mathbb { R } \}.$$
\textbf{Questions}
(31) The plane passing through the origin and not intersecting either of $\ell _ { 1 }$ and $\ell _ { 2 }$ has equation $ax + by + cz = d$. Write the value of $| a + b + c + d |$ where $a, b, c, d$ are integers with $\gcd = 1$.\\
(32) Let $r$ be the smallest possible radius of a circle that has a point on $\ell _ { 1 }$ as well as a point on $\ell _ { 2 }$. It is given that $r ^ { 2 }$ (i.e., the square of the smallest radius) is an integer. Write the value of $r ^ { 2 }$.