Collinearity and Ratio Involving Position Vectors

With respect to the origin, the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively with \(\mathbf{a}\) not parallel to \(\mathbf{b}\).

1. Show that \(A\), \(B\) and \(C\) are collinear given that non-zero numbers \(\lambda \) and \(\mu \) are such that \(\lambda \mathbf{a}+\mu \mathbf{b}+\mathbf{c}=0\); \(\lambda +\mu +1=0\).[2]
2. Given that the point \(P\) with position vector \(\mathbf{p}\) lies on \(BA\) produced such that \(2\mathbf{p}=7\mathbf{a}-5\mathbf{b}\), find the ratio \(AP:BP\).[2]