Animated Solution for Mathematics - Straight Lines: Two sides of a parallelogram are along the lines, $x+y=3$ and $x-y+3=0$. If its diagonals intersect at $(2,4)$, then one of its vertex is :

Welcome, future engineers! Today, we are going to dissect a parallelogram, not just as a shape on a page, but as a system of constraints. In JEE Advanced, geometry is rarely about drawing; it is about understanding the hidden relationships between lines and points.

We are given two lines, $L_1:x+y=3$ and $L_2:x-y=-3$. These are our adjacent sides. Because their slopes are different, they must intersect.

This intersection is the birth of our parallelogram—the vertex $A$. By solving the system:

Adding these equations, we find $2x=0$, which implies $x=0$. Consequently, $y=3$. Thus, our first vertex is $A(0,3)$.

The Diagonal Intersection

Now, we are given the intersection of the diagonals at $M(2,4)$. This is the 'heart' of the parallelogram.

Why is this point so important? Because of the property of symmetry! The diagonals of a parallelogram bisect each other.

This means $M$ is the midpoint of the diagonal connecting $A$ to its opposite vertex, $C$. If we let $C$ be $(x_c,y_c)$, the midpoint formula tells us:

We have found the opposite vertex $C(4,5)$.

Finding the Missing Vertices

We are almost there. We know the opposite sides of a parallelogram are parallel.

The side passing through $C$ must be parallel to $L_1$, so it takes the form $x+y=k$. Substituting $C(4,5)$, we get $4+5=9$, so the line is $x+y=9$.

Similarly, the side through $C$ parallel to $L_2$ is $x-y=c$. Substituting $C(4,5)$, we get $4-5=-1$, so the line is $x-y=-1$.

Now, to find the remaining vertices, we look for the intersection of these new lines with the original ones. Vertex $D$ is the intersection of $L_2$ ($x-y=-3$) and the new line $x+y=9$.

Adding these equations, we get $2x=6$, so $x=3$. Substituting back, $3+y=9$, so $y=6$.

Vertex $D$ is $(3,6)$.

Looking at our options, $(3,6)$ is right there! You have successfully navigated the geometry, applied the midpoint theorem, and solved the system. This is the essence of JEE mathematics—logical, precise, and deeply satisfying.