First slide

Equation of line in Straight lines

Question

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

Moderate

A

$2 a$
B

$a \sqrt{2}$
C

$2 \sqrt{2} a$
D

$a / \sqrt{2}$

Solution

The intercepts between the axes made by the given lines are $a \sqrt{2}, a r \sqrt{2}, a r^{2} \sqrt{2}, \dots$
So the required sum $= \sqrt{2} [a + a r + a r^{2} + \dots]$
$= \sqrt{2} \frac{a}{1 - r} = \frac{\sqrt{2} a}{1 - (1 / 2)} = 2 \sqrt{2} a$

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