Question

The points $\left(a, b + c\right)$, $\left(b, c + a\right)$ and $\left(x, a + b\right)$ are collinear, if x =

• A. $a$
• B. $b$
• C. $c$
• D. $-a$

Answer 1

Answer: (C)

Since the three points are collinear

$\therefore $ Area of triangle = 0

$\Rightarrow \quad\Delta=\frac{1}{2}\left|\begin{matrix}a&b+c&1\\ b&c+a&1\\ x&a+b&1\end{matrix}\right|=0$

$\Rightarrow \quad\left|\begin{matrix}a&b+c&1\\ b&c+a&1\\ x&a+b&1\end{matrix}\right|=0$

Applying $R_{1} \rightarrow R_{1}- R_{2}, R_{2} \rightarrow R_{2}-R_{3}$, we get

$\left|\begin{matrix}a-b&-\left(a-b\right)&0\\ b-x&-\left(b-c\right)&0\\ x&a+c&1\end{matrix}\right|=0$

Expanding along $C_{3}$, we get

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

$\Rightarrow \quad ab-b^{2}-ac+bc=ab-ax-b^{2}+bx$

$\Rightarrow \quad c\left(b-a\right)=x \left(b-a\right)\Rightarrow x=c$