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

Answer: (C)

a = 1 + b

Answer 2

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 $\left(b + 1 - a\right) = 0$