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

Answer 1

The correct option is C $2\sqrt{2}a$ units

$x+y=a$
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, $x+y=a,x+y=ar,\dots x+y=a{r}^{n-1},\cdots$ $=\sqrt{2}a,\sqrt{2}ar,\dots \sqrt{2}a{r}^{n-1},\cdots$
Sum $=\sqrt{2}a+\sqrt{2}ar+\dots +\sqrt{2}a{r}^{n-1}+\dots$
$=\sqrt{2}(a+ar+...a{r}^{n-1}+...)$
Using sum of infinite terms in GP formula we get
$=\sqrt{2}\left(\frac{a}{1-r}\right)$
$=\frac{\sqrt{2}a}{1-r}=2\sqrt{2}a\text{units}(\because r=\frac{1}{2})$