The points $A$ and $B$ lie on a circle with center $O$ and radius $\lambda$ unit. With refer

Question

The points $A$ and $B$ lie on a circle with center $O$ and radius $\lambda$ unit. With reference to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. The point $X$ on the line segment $AB$ is such that $AX:XB = 1:3$ and the point $Y$ is the foot of perpendicular of $X$ on $OB$. Find the position vector of $X$.[1] It is given that the acute angle $AOB$ is $\frac{\pi}{6}$. Find $\mathbf{a} \cdot \mathbf{b}$ in terms of $\lambda$.[1] Show that the position vector of $Y$ is $\frac{2 + 3\sqrt{3}}{8} \mathbf{b}$.[2] Hence find the exact area of $\Delta OXY$ in term of $\lambda$.[4]

Accepted Answer

(a) $\overrightarrow{OX} = \dfrac{3\mathbf{a} + \mathbf{b}}{4}$ (b) $\dfrac{\sqrt{3}}{2}\lambda^2$ (d) $\dfrac{3(2 + 3\sqrt{3})}{128}\lambda^2$ units$^2$

2025 TJC P1 Q6

2025 TJC P1 Q6

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