question_answer

If \[a\ne b\ne 0\], prove that the points \[(a,{{a}^{2}}),(b,{{b}^{2}})(0,0)\] will not be collinear.

Answer:

Area \[=\frac{1}{2}|({{b}^{2}}-{{c}^{2}})+b({{c}^{2}}-{{a}^{2}})+c({{a}^{2}}-{{b}^{2}})|\]

\[=\frac{1}{2}|a(b-c)(b+c)-{{a}^{2}}(b-c)-bc(b-c)|\]

\[=\frac{1}{2}|(b-c)(a(b+c)-{{a}^{2}}-bc|\]

\[=\frac{1}{2}|(b-c)(ab+ac-{{a}^{2}}-bc)|\]

\[=\frac{1}{2}|(b-c)(a-b)(c-a)|\]

this can never be zero as \[a\ne b\ne c\]

Hence these point can never be collinear.               Hence Proved.