In a quadrilateral $\mathrm{PQRS}, \mathrm{M}$ and $\mathrm{N}$ are mid-points of the sides $P Q$ and RS…

In a quadrilateral $\mathrm{PQRS}, \mathrm{M}$ and $\mathrm{N}$ are mid-points of the sides $P Q$ and RS respectively. If $\overline{\mathrm{PS}}+\overline{\mathrm{QR}}=\mathrm{t} \overline{\mathrm{MN}}$, then $\mathrm{t}=$
  1. $\frac{1}{2}$
  2. 4
  3. $\frac{3}{2}$
  4. 2

Solution

$\mathrm{p}, \mathrm{q}, \mathrm{r}, \mathrm{s}, \mathrm{m}, \mathrm{n}$ be the position vectors of points $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}, \mathrm{M}, \mathrm{N}$. Now $\overline{\mathrm{PS}}+\overline{\mathrm{QR}}$ $=\overline{\mathrm{s}}-\overline{\mathrm{p}}+\overline{\mathrm{r}}-\overline{\mathrm{q}}=(\overline{\mathrm{s}}+\overline{\mathrm{r}})-(\overline{\mathrm{p}}+\overline{\mathrm{q}})$ Since $\mathrm{M}$ and $\mathrm{N}$ are mid-points of PQ and RS respectively, we write $\overline{\mathrm{m}}=\frac{\overline{\mathrm{p}}+\overline{\mathrm{q}}}{2}$ and $\overline{\mathrm{n}}=\frac{\overline{\mathrm{r}}+\overline{\mathrm{s}}}{2}$ $\therefore$ eq. (1) becomes $\overline{\mathrm{PS}}+\overline{\mathrm{QR}}=2 \overline{\mathrm{n}}-2 \overline{\mathrm{m}}=2(\overline{\mathrm{n}}-\overline{\mathrm{m}})=2 \overline{\mathrm{MN}}$ From given data, $\mathrm{t}=2$

Asked in: MHT CET 2021 (21 Sep Shift 1)

Practice more Vectors questions on Aicharya