2025 ASRJC P1 Q9

The line \(\ell\) has the equation \(\frac{x-a}{2}=y-2=\frac{1-2z}{b}\), where \(a\) and \(b\) are real constants. The plane \(\pi\) has the equation \(3x-y+4z=10\).

1. Given that the line \(\ell\) and the plane \(\pi\) do not intersect, show that \(a \neq \frac{10}{3}\) and \(b = \frac{5}{2}\).[5]
2. It is given that \(a=1\) and \(b=3\). Find the point of intersection between the line \(\ell\) and the plane \(\pi\).[3]
3. It is given now that \(a=1, b=\frac{5}{2}\).
   1. Find the distance between the line \(\ell\) and the plane \(\pi\).[2]
   2. Determine whether the line \(\ell\) and the origin \(O\) lie on the same side of the plane \(\pi\). State an equation of the other plane that is equidistant from line \(\ell\) and parallel to plane \(\pi\).[3]