Straight LineHardBloom L3
Question
The number of integral points inside the triangle made by the line $3x + 4y - 12 = 0$ with the coordinate axes which are equidistant from at least two sides is/are (an integral point is a point both of whose coordinates are integers):
Options
A.1
B.2
C.3
D.4
Solution
Sol. [Key Concept: Interior lattice points of the $3$–$4$–$5$ triangle equidistant from $\geq 2$ sides lie on the angle bisectors.]
Point $P$ must lie on at least one of the angle bisectors.
Incentre,
$$\begin{matrix} I \equiv \left( \frac{4(0) + 3(4) + 5(0)}{4 + 3 + 5},\frac{4(3) + 3(0) + 5(0)}{4 + 3 + 5} \right) & \\ & I \equiv (1,1) \\ & P \equiv (1,1)\text{~}\text{only}\text{~} \end{matrix}$$
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