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Standard VIII

Mathematics

Total Surface Area of a Cuboid

Prove that th...

Question

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.

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Solution

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 $a d = b c$ .....(1)
Since AC is $y - b = \frac{d}{c} (x - a)$
It will pass through (0, 0) if $- b = \frac{d}{c} (- a)$
or $b c = a d$ which is true by (1).

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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.

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

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 $a d = b c$ .....(1)
Since AC is $y - b = \frac{d}{c} (x - a)$
It will pass through (0, 0) if $- b = \frac{d}{c} (- a)$
or $b c = a d$ which is true by (1).