If , prove that the points (a, a 2 ), (b,b 2 ),(c, c 2 ) can never be collinear.

Question

Philoid · Accepted Answer

Area of the triangle having vertices (x₁,y₁), (x₂,y₂) and (x₃,y₃)

= |x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)|

∴ Area = | a (b² - c²) + b( c² - a²) + c(a² – b²)|

= | ab² - ac² + bc² - ba² + ca² – cb²|

= | (b – c)(- a² ) + ab +ac –bc|

= | (b – c)(a - b)(c – a)|

Also it is given that .

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