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  4. 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

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Q. 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

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Solution:

Equation of $A B$ is $\frac{x}{21}+\frac{y}{21}=1$ or
$x+y=21$
image
Let $(h, k)$ be any pt. inside the $\Delta O A B$ 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
image
$\therefore $ Total no. of integral points
$=19+18+17+\ldots+1$
$=\frac{19 \times 20}{2}=190$