Animated Solution for Mathematics - Probability: If a point $A(x,y)$ lies in the region bounded by the $y$-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is:

Welcome, fellow explorer! Today, we are not just solving a math problem; we are mapping a territory. Imagine you are standing on the Cartesian plane, looking at three lines that carve out a specific, enclosed region.

Our goal is to find the probability that a randomly chosen point within this region has a $y$-coordinate less than $1$. This is a classic JEE Advanced problem that tests your ability to visualize geometry and apply the elegance of complementary probability.

First, let us identify our boundaries. We have the $y$-axis, defined by $x=0$. Then, we have two lines: $L_1:x+2y=6$ and $L_2:5x-6y=30$.

To understand the region, we must find the vertices of the triangle they form. By setting $x=0$ in $L_1$, we find the intersection $(0,3)$. Setting $x=0$ in $L_2$, we find $(0,-5)$.

These two points lie on our $y$-axis, forming the base of our triangle. To find the third vertex, we solve the system of equations for $L_1$ and $L_2$:

Adding these equations eliminates $y$, leaving $8x=48$, so $x=6$. Substituting $x=6$ back into $L_1$ yields $y=0$. Our third vertex is $(6,0)$.

Now that we have our vertices $(0,3)$, $(0,-5)$, and $(6,0)$, the triangle is clear. The base lies on the $y$-axis, spanning from $y=3$ to $y=-5$, giving us a length of $3-(-5)=8$ units.

The height is the perpendicular distance from the third vertex $(6,0)$ to the $y$-axis, which is simply the $x$-coordinate, $6$ units. The total area is calculated as follows:

This is our total sample space.

The problem asks for the probability that $y<1$. If we draw the line $y=1$, it slices through our triangle.

The region where $y<1$ is the larger part of the triangle, while the region where $y\ge 1$ is a smaller triangle at the top. Calculating the area of the larger part directly would involve a quadrilateral, which is tedious.

Instead, we use the power of complementary probability:

Let us focus on the upper triangle where $y\ge 1$. Its vertices are $(0,3)$, $(0,1)$, and the intersection of $y=1$ and $L_1$.

Substituting $y=1$ into $x+2y=6$, we get $x+2=6$, so $x=4$. The third vertex is $(4,1)$.

The base of this smaller triangle lies on the line $y=1$, with length $4-0=4$. The height is the vertical distance from $y=1$ to $y=3$, which is $3-1=2$. The area of this upper triangle is:

The probability $P(y\ge 1)$ is the ratio of the upper triangle's area to the total area:

Finally, the probability that $y<1$ is:

By visualizing the geometry and choosing the path of least resistance, we have arrived at the solution with precision and grace. Keep practicing this mindset—it is the key to mastering JEE Advanced.