Question

If the points A(1,2), B(0,0), and C(a,b) are collinear, then which of the following is correct?

• A. a = b
• B. a = 2b
• C. 2a = b
• D. a = –b

Answer 1

The correct option is C

2a = b

Let the given points be A = $(x_1,y_1)=(1,2)$
$B=(x_2,y_2)=(0,0)$ and $C=(x_3,y_3)=(a,b).$
$\because$ Area of $\Delta$ ABC
$=\Delta =\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]\therefore \Delta =\frac{1}{2}\left[1\right(0-b)+0(b-2)+a(2-0\left)\right]=\frac{1}{2}(-b+0+2a)=\frac{1}{2}(2a-b)$
Since, the points A(1,2), B(0,0) and C(a,b) are collinear, then area of the triangle ABC should be equal to zero.
i.e, area of $\Delta$ ABC = 0
$\Rightarrow \frac{1}{2}(2a-b)=0\Rightarrow 2a-b=0\Rightarrow 2a=b$
Hence, the required relation is 2a = b.