Question

In $\Delta ABC,A=(\alpha ,\beta ),B=(1,2),C=(2,3)$ and point $A$ lies on the line $y=2x+3$, where $\alpha ,\beta$ are integers. Area of the triangle is $S$, such that $\left[S\right]=2$, where $[\cdot ]$ denotes the greatest integer function. Which of these can be possible coordinates of $A$ ?

• A. $(-7,-11)$
• B. $(-6,-9)$
• C. $(2,7)$
• D. $(3,9)$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $(-6,-9)$
B $(2,7)$
C $(-7,-11)$
D $(3,9)$

The equation of line joining $(1,2)$ and $(2,3)$ is $y-x-1=0$
Let $A$ have the coordinates $(h,2h+3)$
The area of $\Delta ABC=\frac{1}{2}BC\times$ height $AD$
Area $=\frac{1}{2}\times \sqrt{2}\times \frac{|2h+3-h-1|}{\sqrt{2}}$
$\Rightarrow$ Area $=\frac{1}{2}\times |h+2|$
We have, $\left[Area\right]=2$
$\Rightarrow 2\le Area<3$
$\Rightarrow 4\le |h+2|<6$
$\therefore -4\ge h+2>-6$ or $4\le h+2<6$
$\therefore -6\ge h>-8$ or $2\le h<4$
Hence, the possible values of $h$ are$-7,-6,2,3$
Hence, the co-ordinates of $A$ can be $(-7,-11),(-6,-9),(2,7),(3,9)$