JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Straight Lines

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Let S be the set of all triangle in the xy - plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in

S

has area 50 sq. units, then the number of elements in the set

S

is:

(A)9
(B)18
(C)32
(D)36✓ Correct

Answer: (D)

Solution

Put the right-angle vertex at the origin and the other two on the axes at

(α, 0)

and

(0, β)

with

α, β

non-zero integers. The area is

\frac{1}{2} ∣ α β ∣ = 50

, so

∣ α β ∣ = 100

. Now

100 = 2^{2} \cdot 5^{2}

has

(2 + 1) (2 + 1) = 9

positive divisors, giving

9

ordered pairs

(∣ α ∣, ∣ β ∣)

of positive factors. Each pair admits four sign choices, one per axis, so the count is

9 \times 4 = 36

triangles.

JEE Main · 2019 · 09 Jan 2019 · Shift 2 · Single correct option

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