Question

Assertion (A) The points $A(a,b+c),B(b,c+a)$ and $C(c,a+b)$ are collinear.
Reason (R) Area of a triangle with three collinear points is zero.

• A. Both A and R are correct; R is the correct explanation of $A$
• B. Both $A$ and $R$ are correct; $R$ is not the correct explanation of $A$
• C. A is correct; R is incorrect
• D. $R$ is correct; $A$ is incorrect

Answer 1

Answer: (A)

We know that, the area of triangle with three collinear points is zero.
Now, consider
$\text{ Area of }△ABC=\frac{1}{2}\left|\begin{array}{ccc}a& b+c& 1\\ b& c+a& 1\\ c& a+b& 1\end{array}\right|$
$=\frac{1}{2}\mid a\{(c+a)\times 1-(a+b)\times 1\}-(b+c)\{b\times 1-1\times c\}$
$+1\{b\times (a+b)-(c+a)\times c\}\mid$
$=\frac{1}{2}\mid a(c+a-a-b)-(b+c)(b-c)$
$+1\left(ab+b^2-c^2-ac\right)\mid$
$=\frac{1}{2}\left|ac-ab-b^2+c^2+ab+b^2-c^2-ac\right|=\frac{1}{2}\times 0=0$
Since, area of $△ABC=0$.
Hence, points $A(a,b+c),B(b,c+a)$ and $C(c,a+b)$ are collinear.