If the points (a, b),(a', b') and (a - a', b | vedclass

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(A) Three points $(x_1, y_1)$,$(x_2, y_2)$,and $(x_3, y_3)$ are collinear if the area of the triangle formed by them is zero,or if the slope between a

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$ab' = a'b$

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(A) Three points $(x_1, y_1)$,$(x_2, y_2)$,and $(x_3, y_3)$ are collinear if the area of the triangle formed by them is zero,or if the slope between any two pairs of points is equal.Let the points be $P(a, b)$,$Q(a', b')$,and $R(a - a', b - b')$.The slope of $PQ$ is $m_1 = \frac{b' - b}{a' - a}$.The slope of $QR$ is $m_2 = \frac{(b - b') - b'}{(a - a') - a'} = \frac{b - 2b'}{a - 2a'}$.Since the points are collinear,$m_1 = m_2$:$\frac{b' - b}{a' - a} = \frac{b - 2b'}{a - 2a'}$$(b' - b)(a - 2a') = (b - 2b')(a' - a)$$ab' - 2a'b' - ab + 2a'b = a'b - ab - 2a'b' + 2ab'$$ab' + 2a'b = a'b + 2ab'$$a'b = ab'$Therefore,$ab' = a'b$.

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

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