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Question

# Let S be the set of points whose abscissas and ordinates are natural numbers. Let P∈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
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B
3
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C
5
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D
11
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Solution

## The correct option is B 3The sum of the distances of P from A(8,0),B(0,12) is minimum if P lies on line joining AB and between themAP+PB=ABAP′+P′B>AB by triangle propertySo let P(x,y)The line AB is x8+y12=1ie 3x+2y=24⇒3x=2(12−y)So y can be 3,6,9y=3,x=6→1 pointy=6,x=4→1 pointy=9,x=2→1 pointSo in total three points.

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