2.6: Quadrilateral with intersecting lines and ratios

0606/22/O/N/20 — Question 9 · 9 marks

2.6 diagram

In the diagram

\overrightarrow{OP} = 2\mathbf{b}

,

\overrightarrow{OS} = 3\mathbf{a}

,

\overrightarrow{SR} = \mathbf{b}

and

\overrightarrow{PQ} = \mathbf{a}

. The lines

OR

and

QS

intersect at

X

.

(a)

Find

\overrightarrow{OQ}

in terms of

\mathbf{a}

and

\mathbf{b}

.

[1]

(b)

Find

\overrightarrow{QS}

in terms of

\mathbf{a}

and

\mathbf{b}

.

[1]

(c)

Given that

\overrightarrow{QX} = \mu\overrightarrow{QS}

, find

\overrightarrow{OX}

in terms of

\mathbf{a}

,

\mathbf{b}

and

\mu

.

[1]

(d)

Given that

\overrightarrow{OX} = \lambda\overrightarrow{OR}

, find

\overrightarrow{OX}

in terms of

\mathbf{a}

,

\mathbf{b}

and

\lambda

.

[1]

(e)

Find the value of

\lambda

and of

\mu

.

[3]

(f)

Find the value of

\dfrac{QX}{XS}

.

[1]

(g)

Find the value of

\dfrac{OR}{OX}

.

[1]