The points (-a,-b), (a, b), (0,0) and (a^{2}, | vedclass

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(A) Let the four points be $A(-a,-b)$,$B(a, b)$,$C(0,0)$,and $D(a^{2}, ab)$.To check if $A, B$,and $C$ are collinear,we calculate the determinant:$\le

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collinear

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(A) Let the four points be $A(-a,-b)$,$B(a, b)$,$C(0,0)$,and $D(a^{2}, ab)$.To check if $A, B$,and $C$ are collinear,we calculate the determinant:$\left|\begin{array}{ccc}-a & -b & 1 \ a & b & 1 \ 0 & 0 & 1\end{array}ight| = -a(b-0) + b(a-0) + 1(0) = -ab + ab = 0$.Since the determinant is $0$,points $A, B$,and $C$ are collinear.Now,check if $B, C$,and $D$ are collinear:$\left|\begin{array}{ccc}a & b & 1 \ 0 & 0 & 1 \ a^{2} & ab & 1\end{array}ight| = a(0-ab) - b(0-a^{2}) + 1(0) = -a^{2}b + a^{2}b = 0$.Since the determinant is $0$,points $B, C$,and $D$ are collinear.Since $A, B, C$ are collinear and $B, C, D$ are collinear,all four points $A, B, C$,and $D$ must lie on the same line.

The points $(-a,-b), (a, b), (0,0)$ and $(a^{2}, ab)$ where $a \neq 0, b \neq 0$ are always

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