Question

The number of intergral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices $(0,0),(0,21)$ and $(21,0)$, is

• A. 133
• B. 190
• C. 233
• D. 105

Answer 1

Answer: (B)

Equation of $AB$ is $\frac{x}{21}+\frac{y}{21}=1$ or
$x+y=21$

Let $(h,k)$ be any pt. inside the $\Delta OAB$ then we must have
$h<21,h>0\dots$(i)
$k<21,k>0\dots$(ii)
and $h+k<21\dots$(iii)
For integral values of $(h,k)$ satisfying (i), (ii) and (iii) simultaneously,
let Total no. of pts

$\therefore$ Total no. of integral points
$=19+18+17+\dots +1$
$=\frac{19\times 20}{2}=190$