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(B) The sum of the distances of a point $P(x, y)$ from $A(8, 0)$ and $B(0, 12)$ is minimized when $P$ lies on the line segment $AB$.The equation of th

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$3$

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(B) The sum of the distances of a point $P(x, y)$ from $A(8, 0)$ and $B(0, 12)$ is minimized when $P$ lies on the line segment $AB$.The equation of the line passing through $(8, 0)$ and $(0, 12)$ is given by $\frac{x}{8} + \frac{y}{12} = 1$.Multiplying by $24$, we get $3x + 2y = 24$.Since $x$ and $y$ must be natural numbers $(x, y \in \mathbb{N})$, we check for integer solutions on the line segment $0 If $x = 2$, $3(2) + 2y = 24$ $\Rightarrow 2y = 18$ $\Rightarrow y = 9$.If $x = 4$, $3(4) + 2y = 24$ $\Rightarrow 2y = 12$ $\Rightarrow y = 6$.If $x = 6$, $3(6) + 2y = 24$ $\Rightarrow 2y = 6$ $\Rightarrow y = 3$.These points $(2, 9), (4, 6), (6, 3)$ are all in $S$.Thus, there are $3$ such points.

Let $S$ be the set of points whose abscissae and ordinates are natural numbers. Let $P \in S$ be such that the sum of the distances 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

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