JEE Main 2023 — Mathematics Coordinate Geometry

Given:

$$3x+4y=60$$3x+4y=60

$$\Rightarrow \frac{x}{20}+\frac{y}{15}=1$$⇒20x​+15y​=1

If $x=1,y=\frac{57}{4}=14.25$x=1,y=457​=14.25

So, points are

$(1,1)(1,2)-(1,14)\Rightarrow 14$(1,1)(1,2)−(1,14)⇒14 points.

If $x=2,y=\frac{27}{2}=13.5$x=2,y=227​=13.5

So, points are

$(2,2)(2,4)\ldots (2,12)\Rightarrow 6$(2,2)(2,4)…(2,12)⇒6 points.

If $x=3,y=\frac{51}{4}=12.75$x=3,y=451​=12.75

So, points are

$(3,3)(3,6)-(3,12)\Rightarrow 4$(3,3)(3,6)−(3,12)⇒4 points.

If $x=4,y=12$x=4,y=12

So, points are

$(4,4)(4,8)\Rightarrow 2$(4,4)(4,8)⇒2 points.

If $x=5,y=\frac{45}{4}=11.25$x=5,y=445​=11.25

So, points are

$(5,5),(5,10)\Rightarrow 2$(5,5),(5,10)⇒2 points.

If $x=6,y=\frac{21}{2}=10.5$x=6,y=221​=10.5

So, point is

$(6,6)\Rightarrow 1$(6,6)⇒1 point

If $x=7,y=\frac{39}{4}=9.75$x=7,y=439​=9.75

So, point is

$(7,7)\Rightarrow 1$(7,7)⇒1 point

If $x=8,y=9$x=8,y=9

So, point is

$(8,8)\Rightarrow$(8,8)⇒ $1$1 point

If $x=9\Rightarrow y=\frac{33}{4}=8.25\Rightarrow$x=9⇒y=433​=8.25⇒ no point

Total points inside the triangle

$=31$=31 points