Question

If $a\ne b\ne c,$ then points $(a,a^2),(b,b^2)$ and $(c,c^2)$ can never be collinear.

• A. True
• B. False

Answer 1

The correct option is A True

If the area of the triangle formed by joining the given points is zero then only the points are collinear.

Area of triangle = $\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$

Here, $x_1=a,y_1=a^2,x_2=b,y_2=b^2,x_3=c,y_3=c^2$

Substituting these values in the formula,

Area of the triangle = $\frac{1}{2}\left[a\right(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2\left)\right]$

$=\frac{1}{2}[a{b}^2-a{c}^2+b{c}^2-a^{2}b+a^{2}c-c{b}^2]$

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

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

$=\frac{1}{2}\left[\right(b-c\left)\right(a-b\left)\right(c-a\left)\right]$

Given that, $a\ne b\ne c$

Therefore, area of the triangle $\ne$ 0.

Hence, the given points can never be collinear.