0606 P11 - Nov 2022 - Q11 - 10 marks
7857
The diagram shows a triangle \(OAC\). The point \(B\) lies on \(AC\) such that \(AB:AC=2:5\). It is given that
\(\overrightarrow{OA}=\mathbf a,\qquad \overrightarrow{OB}=\mathbf b,\qquad \overrightarrow{OC}=\mathbf c.\)
(a) Show that
\(5\mathbf b-3\mathbf a=2\mathbf c.\)
The diagram now includes points \(X\) and \(Y\), such that
\(\overrightarrow{OX}=\frac34\overrightarrow{OA}\)
and
\(\overrightarrow{OY}=m\overrightarrow{OB},\)
where \(m\) is a constant. It is also given that \(XY:XC=\lambda:1\), where \(\lambda\) is a constant.
(b) Using part (a), find \(\overrightarrow{XC}\) in terms of \(\mathbf a\) and \(\mathbf b\).
(c) Hence find the values of \(m\) and \(\lambda\).
