First slide

Cartesian plane

Question

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

Moderate

A

$a - b = 2$
B

$a + b = 2$
C

$a = 1 + b$
D

$a = 1 - b$

Solution

Given points will be collinear, if
$\begin{matrix} |\begin{array}{rrr} a & b & 1 \\ b & a & 1 \\ a^{2} & - b^{2} & 1 \end{array}| = 0 \\ \Rightarrow |\begin{array}{rrr} a & b & 1 \\ b - a & a - b & 0 \\ a^{2} - a & - b^{2} - b & 0 \end{array}| = 0 \end{matrix}$ $\begin{matrix} Applying R_{2} \to R_{2} - R_{1}, \\ R_{3} \to R_{3} - R_{1} \end{matrix}$
$\begin{array}{l} \Rightarrow (a - b) |\begin{array}{rrr} a & b & 1 \\ - 1 & 1 & 0 \\ a^{2} - a & - b^{2} - b & 0 \end{array}| = 0 \\ \Rightarrow (a - b) (b^{2} + b - a^{2} + a) = 0 \\ \Rightarrow (a - b) \{(a + b) - (a^{2} - b^{2})\} = 0 \\ \Rightarrow (a - b) (a + b) (1 - a + b) = 0 \\ \Rightarrow a = b or, a + b = 0 or a = 1 + b \end{array}$

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