Question

One possible condition for the three points $(a, 5), (b, a)$ and $(a^2, - b^2)$ to be collinear is

• A. a - b = 2
• B. a + b = 2
• C. a = 1 + b
• D. a = 1 - b

Answer 1

The Correct Option is C

Since, points $(a, b), (b, a)$ and $(a^2, - b^ 2)$ are collinear.
$\therefore \begin{vmatrix}a&b&1\\ b&a&1\\ a^{2}&-b^{2}&1\end{vmatrix} = 0 $
Applying, $R_{2} \to R_{2} - R_{1}$ and $R_{3} \to R_{3} -R_{1}$ $\Rightarrow\begin{vmatrix}a&b&1\\ b-a&a-b&0\\ a^{2}-a&-b^{2}-b&0\end{vmatrix} = 0$
$\Rightarrow\begin{vmatrix}b-a&a-b\\ a^{2}-a&-b^{2}-b\end{vmatrix} = 0$
$\Rightarrow \left(a-b\right)\begin{vmatrix}-1&1\\ a^{2}-a&-b^{2}-b\end{vmatrix} = 0$
$\Rightarrow \left(a-b\right)\left(b^{2} +b+a^{2}+a\right) = 0 $
$\Rightarrow \left(a-b\right)\left(a+b\right)\left(b+1-a\right) =0 $
$\Rightarrow $ Either $a - b = 0$ or $\left(a + b\right) = 0$
or $\left(b + 1 - a\right) = 0$