Let be the set of all triangles in the -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in has area 50 sq. units, then the number of elements in the set is
- A.36
- B.32
- C.18
- D.9
Show answer and solution
Correct option: A
Solution
The triangle has one vertex at origin, one on the x-axis at (a, 0), and one on the y-axis at (0, b). Area = (1/2)|a||b| = 50, so |a||b| = 100.
We need integer pairs (a, b) with |a|·|b| = 100. The number of positive divisor pairs (|a|, |b|) with |a|·|b| = 100: since 100 = 2²·5², it has (2+1)(2+1) = 9 positive divisors, giving 9 ordered pairs (|a|, |b|).
Each of |a| and |b| can independently be positive or negative, giving 2×2 = 4 sign combinations per divisor pair. Total = 9 × 4 = 36.
The answer is 36.
We need integer pairs (a, b) with |a|·|b| = 100. The number of positive divisor pairs (|a|, |b|) with |a|·|b| = 100: since 100 = 2²·5², it has (2+1)(2+1) = 9 positive divisors, giving 9 ordered pairs (|a|, |b|).
Each of |a| and |b| can independently be positive or negative, giving 2×2 = 4 sign combinations per divisor pair. Total = 9 × 4 = 36.
The answer is 36.
[JEE Main 2019 · 9 Jan · Shift II]