If the points $(a, b), (a', b')$ and $(a - a', b - b')$ are collinear, then

$a b ’ = a ’ b$
$a b = a ’ b ’$
$a a ’ = b b ’$
$a^{2} + b^{2}$

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Solution

The correct option is A
$a b ’ = a ’ b$

Step1: Apply the condition of collinearity of three points
We know that three points $X, Y,$ and $Z$ are collinear points if
Slope of $XY =$ Slope of $YZ =$ Slope of $XZ$
Given points are $(a, b), (a', b')$ and $(a - a', b - b')$
$∴ \frac{(b' - b)}{(a' - a)} = \frac{(b - b' - b')}{(a - a' - a')}$ $[∵$ The slope joining the points $(x_{1}, y_{1}), (x_{2}, y_{2})$ is $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $]$
Step 2: Simplify the above equation
$\Rightarrow$ $\frac{(a - a' - a')}{(a' - a)} = \frac{(b - b' - b')}{(b' - b)}$
$\Rightarrow$ $\frac{(a - 2 a')}{(a' - a)} = \frac{(b - 2 b')}{(b' - b)}$
$\Rightarrow$ $\frac{(a - a') - a'}{(a' - a)} = \frac{(b - b') - b'}{(b' - b)}$
$\Rightarrow$ $- 1 - \frac{a'}{(a' - a)} = - 1 - \frac{b'}{(b' - b)}$
$\Rightarrow$ $a' (b' - b) = b' (a' - a)$
$\Rightarrow$ $a' b' - a' b = b' a' - b' a$
$\Rightarrow$ $a' b = b' a$ $[∵ a' b' = b' a']$
Hence, option A is the correct option.

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