Question

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

• A. $\mathrm{a}–\mathrm{b}=2$
• B. $\mathrm{a}+\mathrm{b}=2$
• C. $\mathrm{a}=1+\mathrm{b}$
• D. $\mathrm{a}=\; 1\; –\mathrm{b}$

Answer 1

The correct option is C

$\mathrm{a}=1+\mathrm{b}$

Explanation for the correct answer:

Given points, $(\mathrm{a},\mathrm{b}),(\mathrm{b},\mathrm{a}),(a^2,-b^2)$.are collinear.

Therefore,

$\left|\begin{array}{ccc}a& b& a\\ b& a& 1\\ a^2& -b^2& 1\end{array}\right|=0$

On applying row operations $\mathrm{R}2\to \mathrm{R}2–\mathrm{R}1$ and $\mathrm{R}3\to \mathrm{R}3–\mathrm{R}1$, we get

$(\mathrm{a}-\mathrm{b})(\mathrm{a}+\mathrm{b})(\mathrm{b}+1-\mathrm{a})=0$

So therefore, either $(\mathrm{a}-\mathrm{b})=0,(\mathrm{a}+\mathrm{b})=0\mathrm{or}(\mathrm{b}+1-\mathrm{a})=0.$

$$\begin{gathered} (\mathrm{b}+1-\mathrm{a})=0 \\ \mathrm{a}=1+\mathrm{b}. \end{gathered}$$

Therefore, the correct answer is option (C).