Question

If a ≠ b ≠ c, prove that the points (a, a²), (b, b²), (c, c²) can never be collinear.

Answer 1

GIVEN: If `a≠ b≠ c`

TO PROVE: that the points (a, a²), (b, b²), (c, c²), can never be collinear.

PROOF:

We know three points (x₁, y₁), (x₂, y₂) and (x₃, y₃) are collinear when 

`1/2[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]=0` 

Now taking three point (a, a²), (b, b²), (c, c²), 

Area `=1/2[a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)` 

`=1/2[ab^2-ac^2+bc^2+ba^2+ca^2+cb^2]` 

`=1/2[(a^2c-a^2b)+(ab2-ac^2)+(bc2-b^2c)]` 

`=1/2[-a^2(b-c))+(a(b^2-c^2))-(bc(b-c))]`

`=1/2[(b-c)(-a^2)+(a(b+c))-bc]` 

`=1/2[(b-c)(-a^2)+ab+ac-bc]` 

`=1/2[(b-c)(-a)(a-b)+c(a-b)]` 

`=1/2(b-c)(a-b)(c-a)`

Also it is given that

a ≠ b ≠ c

Hence area of triangle made by these points is never zero. Hence given points are never collinear.