Question

The sum of the intercepts cut off by the axes on the lines $x+y=a,x+y=ar,x+y=a{r}^2\dots \dots \dots$ where $a\ne 0$ and $r=\frac{1}{2}$ is

• A. $2a$
• B. $a\sqrt{2}$
• C. $2\sqrt{2}a$
• D. $\frac{a}{\sqrt{2}}$

Answer 1

Answer: (C)

$2\sqrt{2}a$

$x+y=a$

intercept cut off by the axes $=\sqrt{a^2+a^2}=\sqrt{2}a$

$\therefore$ sum of all intercepts cut off by the axes on the lines,

$x+y=a,x+y=ar,....x+y=a^{r^n-1}$

$\sqrt{2}a,\sqrt{2}ar,....\sqrt{2}a{r}^{n-1}$

$Sum=\sqrt{2}a+\sqrt{2}ar+....+\sqrt{2}a{r}^{n-1}....$as

$=\sqrt{2}\left(\frac{a}{1-r}\right)$

$=\frac{\sqrt{2}a}{1-r}$

Given $\Rightarrow r=\frac{1}{2}$

$\therefore Sum=2\sqrt{2}a$