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  4. The condition on a and b, such that the portion of the line ax + by - 1 = 0 intercepted between the lines a x + y = 0 and x + by = 0 subtends a right angle at the origin, is

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Q. The condition on $a$ and $b$, such that the portion of the line $ax + by - 1 = 0$ intercepted between the lines $a x + y = 0$ and $x + by = 0$ subtends a right angle at the origin, is

Straight Lines

Solution:

Lines $a x + y = 0$ and $x + b y = 0$ intersect at $O (0 , 0)$.
image
Hence, if $A B$ subtends right angle at $O (0 , 0)$, then $ax + y = 0$ and $x + by = 0$ are perpendicular to each other. So,
$(-a)(-\frac{1}{b}) = -1$
or $ a + b = 0$