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Prove that the points ( ab, ) ,( { a }_{ 1 },{ b }_{ 1 } ) and

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Prove that the points $(ab,), (a_{1}, b_{1})$ and $(a - a_{1}, b - b_{2})$ are collinear if $a b_{1} = a_{1} b$

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Prove that the points

(ab,), (a_{1}, b_{1})

and

(a - a_{1}, b - b_{2})

are collinear if

a b_{1} = a_{1} b

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Text solutionVerified

Consider the following points

A (a, b), B (a 1, b 1), C (a - a 1, b - b 1)

Since the given points are collinear, we have

a re a (△ A BC) = 0

First find the area of

a re a (△ A BC)

as follows:

a re a (△ A BC) = \frac{1}{2} ∣ x_{1} (y_{1} - y_{3}) + x 1 (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

= ∣ a (b_{1} - (b - b_{1})) + a_{1} ((b - b_{1}) - b) + (a - a_{1}) (b - b_{1}) ∣

= ∣ a (b_{1} - b + b_{1}) + a_{1} (b - b_{1} - b) + a (b - b_{1}) - a 1 (b - b_{1}) ∣

= ∣ - ab - a_{1} b_{1} + ab - a b_{1} + a_{1} b + a_{1} b_{1} ∣

= ∣ - (a b_{1} - a_{1} b) ∣

= (a b_{1} - a_{1} b)

This gives,

a b_{1} - a_{1} b = 0

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Question Text	Prove that the points $(ab,), (a_{1}, b_{1})$ and $(a - a_{1}, b - b_{2})$ are collinear if $a b_{1} = a_{1} b$
Answer Type	Text solution:1
Upvotes	150