Prove that the points (a, b), (c, d) and (a – c, b – d) are collinear if ad = bc. Also show that the straight line passing through these points passes through origin.

Engineering

Mathematics

Introduction to Determinants

Question

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If the given points be A, B, C then as they are collinear therefore slopes of AC and BC are same.
$∴ \frac{d}{c} = \frac{b}{a}$ or ad = bc .....(1)
Since AC is $y - b = \frac{d}{c} (x - a)$
It will pass through (0, 0) if $- b = \frac{d}{c} (- a)$
or bc = ad which is true by (1).

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