Question

The point $(-a,-b),(0,0),(a,b)$ and $(a^2,ab)$ are-

• A. collinear
• B. concyclic
• C. vertices of a rectangle
• D. vertices of a parallelogram

Answer 1

The correct option is A collinear

The equation of line passing through points $\left(0,0\right)$ and $\left(a,b\right)$ is given by

$\begin{array}{c}L:\left(y-0\right)=\frac{b-0}{a-0}\left(x-0\right)\\ \Rightarrow y=\frac{b}{a}x\\ \Rightarrow ay=bx\\ \Rightarrow L:bx-ay=0\end{array}$

Putting $\left(-a,-b\right)$ in $R.H.S$

we get

$L:-ab-a\left(-b\right)$ Satisfied $L$ $\left(-a,-b\right)$ lies on $L$

Now Putting $\left(a^2,ab\right)$ in $R.H.S$

we get

$L:b\left(a^2\right)-a\left(ab\right)-a^{2}b=0$

Therefore, $\left(a^2,ab\right)$ Satisfies $L$

So $\left(a^2,ab\right)$ lies on $L$

Hence The points $\left(0,0\right),\left(a,b\right),\left(-a,-b\right),\left(a^2,ab\right)$ are Collinear. So the option $\left(A\right)$ is the correct answer.