Nice application

Probability Level 4

Consider the set of all triangles OPQ where O is the origin and P and Q are distinct points in the plane with non- negative integral coordinates (x , y) such that 5x + y = 99. Number of such distint triangles whose area is positive integer is


The answer is 90.

U Z by U Z , 24, Guyana

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1 solution

Pranjal Jain
Dec 10, 2014

There are 20 non-negative integral solutions to 5 x + y = 99 5x+y=99 [(0,99),(1,94),...,(19,4)]

Now ∆ O P Q = 1 2 × P Q × Perpendicular distance of PQ from O ∆OPQ=\dfrac{1}{2}×PQ×\text{Perpendicular distance of PQ from O}

Perpendicular distance = ∣ 5 ( 0 ) + 1 ( 0 ) − 99 5 2 + 1 2 ∣ = 99 26 \text{Perpendicular distance}=\big |\dfrac{5(0)+1(0)-99}{\sqrt{5^{2}+1^{2}}}\big |=\dfrac{99}{\sqrt{26}}

So ∆ O P Q = 99 2 26 × P Q ∆OPQ=\dfrac{99}{2\sqrt{26}}×PQ

Now P Q PQ must be integer multiple of 2 26 2\sqrt{26}

Distance between consecutive integral points on 5 x + y = 99 5x+y=99 comes out to be 26 \sqrt{26} . So alternate points may be selected on line to get integral area. We have two choices now.

  • Both P P and Q Q with even ordinates

Number of cases= ( 10 2 ) = 45 \binom{10}{2}=45

  • Both P and Q with odd ordinates

Number of cases= ( 10 2 ) = 45 \binom{10}{2}=45

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