Question

If a is not equal to b and b is not equal to c . prove that (a,a2) , (b,b2) and (c,c2) can never be collinear

Answer 1

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Answer 2

(a , a² ) , (b , b²)  & (c , c²)  can not be collinear if a≠b  b≠c and a≠c

Step-by-step explanation:

a≠b  b≠c and a≠c

(a , a² ) , (b , b²)  & (c , c²)  are collinear if

area = 0

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

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

iff  a(b + c)(b - c)  +bc² - ba² + ca² - cb² = 0

iff  (ab + ac)(b - c) - bc(b - c) -a²(b - c) = 0

iff (b - c) ( ab + ac - bc - a²) = 0

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

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

iff either a = b   or a = c  or b = c

but given that a≠b  b≠c and a≠c

hence (a , a² ) , (b , b²)  & (c , c²)  can not be collinear if a≠b  b≠c and a≠c

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