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

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

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

Applying $R_1\to R_1-R_2,R_2\to R_2-R_3$, we get

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

Expanding along $C_3$, we get

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

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

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