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 5 x+y=99. Number of such distinct triangles whose area is a positive integer, is

Question

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 5 x+y=99. Number of such distinct triangles whose area is a positive integer, is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Area =(1/2)|0 0 1 x1 99-x1 1 x2 99-x2 1| =(1/2)[x1(99-x2)-x2(99-x1)] text Area = (1/2)|[(x1-x2) 99]| Area is an integer then both x1 and x2 are simultaneously either even or both odd.

hence 10 C 2+ 10 C 2=2 ⋅ 10 C 2=90

Q. 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 distinct triangles whose area is a positive integer, is

Area =21​∣∣​0x1​x2​​099−x1​99−x2​​111​∣∣​ =21​[x1​(99−x2​)−x2​(99−x1​)] Area =21​∣[(x1​−x2​)99]∣ Area is an integer then both x1​ and x2​ are simultaneously either even or both odd. hence 10C2​+10C2​=2⋅10C2​=90

Q. 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 $5 x + y = 99$ . Number of such distinct triangles whose area is a positive integer, is