Straight Lines
Lattice points inside a triangle
nta_pyq_2023_jan
Grade 11
Question:
A triangle is formed by X-axis, Y-axis and the line $3x + 4y = 60$. Then the number of points $P(a, b)$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is ______.
Step-by-Step Solution
Key Concept: The triangle has vertices $(0,0)$, $(20,0)$, $(0,15)$. For strictly interior points: $a > 0$, $b > 0$, $3a + 4b < 60$, with $b = ka$ for some positive integer $k$.
Total valid interior lattice points with $b$ a positive multiple of $a$: 31.
Correct Answer: 31