Question

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ}=a\hat{i}+b\hat{j}$ and $\overrightarrow{PS}=a\hat{i}-b\hat{j}$ are adjacent sides of a parallelogram $PQRS$. Let $\vec{u}$ and $\vec{v}$ be the projection vectors of $\vec{w}=\hat{i}+\hat{j}$ along $\overrightarrow{PQ}$ and $\overrightarrow{PS}$, respectively. If $|\vec{u}|+|\vec{v}|=|\vec{w}|$ and if the area of the parallelogram $PQRS$ is $8$, then which of the following statements is/are TRUE?

• A. $a+b=4$
• B. $a-b=2$
• C. The length of the diagonal $PR$ of the parallelogram $PQRS$ is $4$
• D. $\vec{w}$ is an angle bisector of the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PS}$

Answer 1

Answer: (A), (C)

$\vec{u}=\frac{(a\hat{i}+b\hat{j})\cdot (\hat{i}+\hat{j})\widehat{PQ}}{\sqrt{a^2+b^2}},\vec{v}=\frac{(ai-b\hat{j})\cdot (\hat{i}+\hat{j})\widehat{PS}}{\sqrt{a^2+b^2}}$
Given $|\vec{u}|+|\vec{v}|=|\vec{w}|$
$\Rightarrow |a+b|+|a-b|=\sqrt{2\left(a^2+b^2\right)}$
$=(2a{)}^2=2\left(a^2+b^2\right)$
$a^2=b^2$
$a\ge b$ or $a\le b$
Area $a^2|(\hat{i}+\hat{j})\times (\hat{i}-\hat{j})|=8,a=2$
Diagonal $4i$ or $4j$