Animated Solution for Mathematics - Straight Lines: Area of the triangle formed by the line $x+y=3$ and angle bisectors of the pair of straight lines $x^2-y^2+2y=1$ is

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a problem; we are peeling back the layers of a geometric mystery.

We are given the equation $x^2-y^2+2y=1$ and asked to find the area of a triangle formed by its angle bisectors and the line $x+y=3$. It sounds daunting, but there is an elegance here that will make your heart sing once we uncover it.

At first glance, $x^2-y^2+2y=1$ looks like a hyperbola. Let us rearrange the terms to see the underlying structure:

By adding $1$ to both sides, we complete the square for the $y$ terms:

This is the beauty of algebra! We have transformed a quadratic into a difference of squares using the identity $a^2-b^2=(a-b)(a+b)$. Thus, our "pair of lines" are actually two simple linear equations:

Now, we determine the angle bisectors. The formula for the bisector of two lines $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ is:

Substituting our lines into this formula, we get:

Notice how the $\sqrt{2}$ terms cancel out perfectly. We are left with the simplified relation:

Solving for the positive case yields $y=1$, and solving for the negative case yields $x=0$. These are our bisectors—a horizontal line and a vertical line. They are perpendicular, and they are beautiful.

We are now standing on the threshold of the solution. We have our two bisectors, $x=0$ and $y=1$, and our third line, $x+y=3$. To find the area of the triangle, we identify the vertices:

1. The intersection of $x=0$ and $y=1$ is $A(0,1)$. 2. The intersection of $x=0$ and $x+y=3$ is $B(0,3)$. 3. The intersection of $y=1$ and $x+y=3$ is $C(2,1)$.

Because $x=0$ and $y=1$ are perpendicular, our triangle is right-angled at $A$. The base $AB$ lies along the $y$-axis with length $|3-1|=2$, and the height $AC$ lies along the line $y=1$ with length $|2-0|=2$.

The area of a right-angled triangle is given by:

Plugging in our calculated values:

We have arrived. The final answer is 2 sq. units. You have learned that even the most intimidating equations are often just simple structures in disguise. Keep this perspective, keep practicing, and remember: the beauty of mathematics lies in the clarity we find at the end of the journey.