question_answer

The points \[(-a,\,-b),\ (0,\,0),\ (a\,,b)\]and \[({{a}^{2}},ab)\]are [IIT 1979]

A) Collinear

B) Vertices of a rectangle

C) Vertices of a parallelogram

D) None of these

Correct Answer: A

Solution :

Here area of quadrilateral is equal to area of \[\Delta ABD\]+area of \[\Delta BCD\] \[=\left| \begin{array}{*{35}{r}}    -a & -b & 1  \\    0 & 0 & 1  \\    {{a}^{2}} & ab & 1  \\ \end{array}\, \right|+\left| \begin{array}{*{35}{r}}    0 & 0 & 1  \\    a & b & 1  \\    {{a}^{2}} & ab & 1  \\ \end{array}\, \right|=0\] Hence the points are collinear.