Question

In a trapezium, vector $\overrightarrow{BC}=\alpha \overrightarrow{AD}$. Also, $\overrightarrow{p}=\overrightarrow{AC}+\overrightarrow{BD}$ is collinear with $\overrightarrow{AD}$ and $\overrightarrow{p}=\mu \overrightarrow{AD}$, then which of the following is true?

• A. $\mu =\alpha +2$
• B. $\mu +\alpha =1$
• C. $\alpha =\mu +1$
• D. $\mu =\alpha +1$

Answer 1

The correct option is D $\mu =\alpha +1$

Let $A$ be the origin.
$\overrightarrow{B}C=\overrightarrow{c}-\overrightarrow{b}=\alpha \overrightarrow{d}$ and $\overrightarrow{p}=\overrightarrow{AC}+\overrightarrow{BD}=\mu AD$
Hence, $\overrightarrow{p}=\overrightarrow{c}+\overrightarrow{d}-\overrightarrow{b}=\mu \overrightarrow{d}$
Using $\overrightarrow{c}-\overrightarrow{b}=\alpha \overrightarrow{d}$
$\Rightarrow \alpha +1=\mu$