Question

Let $S$ be the set of points whose abscissas and ordinates are natural numbers. Let $P\in S$ such that the sum of the distance of $P$ from $(8,0)$ and $(0,12)$ is minimum among all elements in $S$. Then the number of such points $P$ in $S$ is

• A. $1$
• B. $3$
• C. $5$
• D. $11$

Answer 1

The correct option is B $3$

As we know, distance is minimum, if points are collinear
$\therefore (x,y),(8,0),(0,12)$ are collinear


$x(-12)+8(12-y)=0$
$\Rightarrow -12x+96-8y=0$
$\Rightarrow 3x+2y-24=0$
$\Rightarrow y=12-\frac{3}{2}x$
As $x$ and $y$ are natural number. Thus, possibles solutions are $(2,9),(4,6),(6,3)$
$\therefore 3$ points are possible.