The Geometry of Homogeneous Systems
My dear student, welcome to a beautiful intersection of algebra and geometry. When you look at the system ax+by+cz=0, bx+cy+az=0, and cx+ay+bz=0, I want you to stop seeing just variables and coefficients.
I want you to see three planes. Because these are homogeneous equations—meaning the constant term on the right side is zero—every single one of these planes is anchored at the origin (0,0,0).
The question before us is: how do these three planes interact? Do they meet at a single point, share a line, or are they the same plane entirely? Let us peel back the layers of this problem together.
The Cyclic Determinant
To understand the intersection, we must look at the coefficient matrix. The determinant of this matrix, which we call Δ, is the heartbeat of the system.
It is defined as:
This is a classic cyclic determinant. If you expand this, you will find it equals 3abc−a3−b3−c3.
Now, here is where the magic happens. We can factorize this expression into a very elegant form:
Δ=−(a+b+c)(a2+b2+c2−ab−bc−ca)
This factorization is the key that unlocks every single condition in our matrix match.
The Sum of Squares
Look closely at that second factor: a2+b2+c2−ab−bc−ca. In the world of JEE Advanced, this is a famous expression.
We can rewrite it by multiplying and dividing by 2, giving us:
21[(a−b)2+(b−c)2+(c−a)2]
Why do we do this? Because a sum of squares is a powerful tool. It tells us that this expression can only be zero if each individual term is zero.
That is, a=b, b=c, and c=a. This means the expression is zero if and only if a=b=c. This realization is the bridge between the algebra and the geometry of the planes.
Analyzing the Conditions
Now, let us walk through the conditions. In Condition (A), we are told $a + b + c
eq 0$ and a2+b2+c2=ab+bc+ca.
The second part forces a=b=c. If we substitute a=b=c into our original equations, they all collapse into a(x+y+z)=0, or simply x+y+z=0.
Geometrically, this means all three planes have become identical. They are perfectly overlapping!
In Condition (B), we have a+b+c=0 but the second factor is non-zero. Since the first factor is zero, the determinant Δ is zero.
This implies the system has infinitely many solutions. By symmetry, if we test the line x=y=z, we find it satisfies all equations. Thus, the planes intersect along this line.
In Condition (C), neither factor is zero, so $\Delta
eq 0$. For a homogeneous system, a non-zero determinant is the hallmark of a unique solution: the trivial solution (0,0,0).
The planes meet only at the origin.
Finally, in Condition (D), both factors are zero. This forces a=b=c AND a+b+c=0, which implies a=b=c=0.
If all coefficients are zero, the equations become 0=0. This is true for every point in the universe! The planes have expanded to fill the entire three-dimensional space.
Conclusion
Mathematics is not about memorizing formulas; it is about recognizing patterns. By understanding the cyclic determinant and the sum of squares, we have transformed a daunting system of equations into a clear geometric story.
Keep practicing this intuition, and you will find that even the most complex problems become simple, elegant truths.
