Question

If a is not equal to b is not equal to c then prove that the points a a square b b square and c c square can never be collinear

Answer 1

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Answer 2

(a , a²) ,  (b , b²)  (c , c²) are never coliinear if  a ≠ c  or b ≠ c  or b ≠ a

Step-by-step explanation:

(a , a²) ,  (b , b²)  (c , c²) are coliinear if

Area = 0

iff (1/2) | a ( b² - c²) + b(c² - a²) + c(a² - b²) | = 0

iff  a ( b² - c²) + b(c² - a²) + c(a² - b²)  = 0

iff   ab² - ac² + bc² - ba² + ca² - cb² = 0

iff   b²(a - c) + ac(a - c) - b(a² - c²) = 0

iff (a - c)(b² + ac) - b(a + c)(a - c) = 0

iff (a - c) (b² + ac - ba - bc) = 0

iff (a - c) ( b(b - a) - c(b - a) ) = 0

iff (a - c)(b - c)(b - a) = 0

iff a = c  or b = c  or b = a

Hence

(a , a²) ,  (b , b²)  (c , c²) are never coliinear if  a ≠ c  or b ≠ c  or b ≠ a

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