There are three distinct points $A$, $B$ and $P$. The point $P$ lies on the circle with centre $O$ a

Question

There are three distinct points $A$, $B$ and $P$. The point $P$ lies on the circle with centre $O$ and diameter $AB$. Relative to the origin $O$, the position vectors of points $A$ and $P$ are $\mathbf{a}$ and $\mathbf{p}$ respectively. By expressing $\overrightarrow{AP}$ and $\overrightarrow{BP}$ in terms of $\mathbf{a}$ and $\mathbf{p}$, show that $\overrightarrow{AP}\cdot \overrightarrow{BP}=0$. [3] The point $C$ with position vector $\mathbf{c}$ lies on $AB$ produced such that $AC:AB=\lambda :1$. Find $\mathbf{c}$ in terms of $\lambda $ and $\mathbf{a}$. [2] It is given that $PC$ is a tangent to the circle and that angle $AOP={{120}^{{}^\circ }}$. Using a suitable scalar product, find the value of $\lambda $. [4]

Accepted Answer

(b) $\mathbf c=(1-2\lambda)\mathbf a$ (c) $\lambda=\frac32$

2025 EJC P1 Q6

2025 EJC P1 Q6

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