A triangle is formed by the X -axis, Y -axis and the line 3 x + 4 y = 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 ______ . [2023]

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Solution

Q.
A triangle is formed by the $X$ -axis, $Y$ -axis and the line $3 x + 4 y = 60 .$ 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 ______ . [2023]

Ans.
(31)

$For point P (a, b) to lie strictly inside the triangle 3 a + 4 b < 60$

Also $a$ is an integer and $b$ is a multiple of $a$

$∴$ $b = n a \Rightarrow 3 a + 4 n a < 60$

$a (3 + 4 n) < 60$

When $a = 1,$ $3 + 4 n < 60,$ $n can take values 1, \dots, 14$

$When a = 2, 2 (3 + 4 n) < 60 \Rightarrow n can take values 1, 2, \dots, 6$ $i.e. 6 point$

$When a = 3, 3 (3 + 4 n) < 60 \Rightarrow n can take values 1, 2, \dots, 4 i.e. 4 point$

$When a = 4, 4 (3 + 4 n) < 60 \Rightarrow n can take values 1, 2 i.e. 2 point$

$When a = 5, 5 (3 + 4 n) < 60 \Rightarrow n can take values 1, 2 i.e. 2 point$

$When a = 6, 6 (3 + 4 n) < 60 \Rightarrow 1 point$

$When a = 7, 7 (3 + 4 n) < 60 \Rightarrow 1 point$

$When a = 8, 8 (3 + 4 n) < 60 \Rightarrow 1 point$

$Total number of points = 14 + 6 + 4 + 2 + 2 + 1 + 1 + 1 = 31$

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