Question

Consider the set of all triangle OPQ where 'O' is origin and P, Q are distinct points in the plane with non-negative integral co-ordinates (x,y) such that 5x+y=99, then the number of such distinct triangle whose area is positive integer are ______​

Answer 1

$\sf \: we \: have \: 20 \: such \: \\ pairs \: which \: lie \: on \: the \: number \: line \: 5x + y = 99 \\ \sf \: now \: distance \: origin \: (0.0) \: from \: this \: line \: is \: d = \frac{99}{ \sqrt{26} } \\ \sf \: let \: p(x1.y1) \: and \: q(x2.y2) \: be \: two \: points \: on \: the \: line \: \\ = > \sf \: pq = \sqrt{(x2 - x1) {}^{2} + (y2 - y1) {}^{2} } \\ \sf \: using \: y2 = 99 - 5x \: we \: get \: pq = |x2 - x1| \\ \sqrt{ 26} \: and \: area \: of \triangle \: opq = \frac{99}{2 \sqrt{26} } |x2 - x1| \sqrt{26} \\ = \frac{99}{2} | x2 - x1| \\ \bf \: for \: area \: \\ = > either \: x2.x. {\bold{e}} \: ({0.2.4.6...18}) \: or \: x2.x. \: e \: (1.3.5.7.19 ) \\ \bf \: number \: of \: such \: possibilities \: = 2x {}^{10} c2 = 2 \times 45 = 90 \\ { \green{ \mathbb{ \bold{answer = 90}}}}$