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)

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