Question

The midpoint of $A\left(0,0\right)$ and $B\left(1024 , \, 2048\right)$ is $A_{1}$ , midpoint of $A_{1}$ and $B$ is $A_{2}$ , midpoint of $A_{2}$ and $B$ is $A_{3}$ and so on. The coordinates of $A_{10}$ are

• A. $\left(1025 , \, 2050\right)$
• B. $\left(1022 , \, 2044\right)$
• C. $\left(1023 , \, 2046\right)$
• D. None of these

Answer 1

Answer: (C)

$A_{1}B=\frac{A B}{2}, \, A_{2}B=\frac{A B}{4}$
$A_{3}B=\frac{A B}{8},........, \, A_{10}B=\frac{A B}{2^{10}}=\frac{A B}{1024}$
$\Rightarrow A \, A_{10}=AB-\frac{A B}{1024}=\frac{1023 \left(A B\right)}{1024}$
$\Rightarrow \, A_{10}$ divides the line joining $A$ and $B$ internally in the ratio $1023:1$
$\Rightarrow \, A_{10}=\left(\frac{0 \times 1 + 1024 \times 1023}{1023 + 1} , \frac{0 \times 1 + 2048 \times 1023}{1023 + 1}\right)$
$=\left(1023 , \, 2046\right)$