Question

Let $O$ be the origin of the coordinate system in the Cartesian plane, $\overline{OP}$ and $\overline{OR}$ be vectors making angles ${45}^o$ and ${135}^o$ respectively with the positive directions of X-axis (i.e., in the counter clock wise). Rectangle OPQR is completed and $M$ is the mid-point of $PQ$. If the line $\overline{OM}$ meets the diagonal $\overline{PR}$ at $T$, and $\overline{OP}=3,\overline{OR}=4$ then $\overline{OT}$ is

• A. $\frac{1}{2}(\overline{i}+\overline{j})$
• B. $\frac{2}{3}(\overline{i}+5\overline{j})$
• C. $\frac{\sqrt{2}}{3}(\overline{i}-5\overline{j})$
• D. $\frac{\sqrt{2}}{3}(\overline{i}+5\overline{j})$

Answer 1

The correct option is D $\frac{\sqrt{2}}{3}(\overline{i}+5\overline{j})$

In $△MTP$ and $△RTO$,

$\angle MTP=\angle RTO$ (vertically opposite angles)

$\angle TMP=\angle TOR$ (alternate angles)

$\therefore$ By AAA criteria,

$△MTP\sim △OTR$

$\frac{MP}{RO}=\frac{PT}{RT}=\frac{1}{2}$

position vector of $\overrightarrow{P}=(\frac{3}{\sqrt{2}},\frac{3}{\sqrt{2}})$

position vector of $\overrightarrow{r}=(\frac{-4}{\sqrt{2}},\frac{4}{\sqrt{2}})$

$\therefore$ By section formula,

Position vector of $\overrightarrow{t}=(\frac{\frac{-4}{\sqrt{2}}+\frac{3}{\sqrt{2}}\times 2}{3},\frac{\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}}\times 2}{3})$

position vector of $\overrightarrow{t}=(\frac{\sqrt{2}}{3},\frac{5\sqrt{2}}{3})$

$\therefore \overrightarrow{OT}=\frac{\sqrt{2}}{3}\hat{i}+\frac{5\sqrt{2}}{3}\hat{j}$