Animated Solution for Mathematics - Complex Numbers: If $z=x+iy,xy\ne 0$, satisfies the equation $z^2+i\bar{z}=0$, then $|z{|}^2$ is equal to :

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving an equation; we are exploring the beautiful, hidden symmetry of the complex plane.

We are given the equation $z^2+i\bar{z}=0$, with the vital constraint that $xy eq 0$. This constraint is not just a footnote; it is a signpost telling us that our solution cannot lie on the axes. Let us embark on this journey to find $|z{|}^2$.

When you face an equation like $z^2+i\bar{z}=0$, your first instinct might be to expand $z$ into $x+iy$. While that is a valid path, it is often a path through a dense forest of algebra.

Instead, let us look for a more elegant route. We want to isolate the terms to analyze their magnitudes. By moving the term $i\bar{z}$ to the right side, we get:

This simple act of balancing the equation is the first step toward clarity. We have now separated the complex variables, setting the stage for a powerful transformation.

Now, we arrive at the most critical moment of our journey. Whenever you see an equation containing both $z$ and its conjugate $\bar{z}$, taking the modulus on both sides is a transformative tool.

It acts like a lens, focusing our attention on the distance from the origin rather than the specific coordinates. Applying the modulus, we get:

This step is profound because it converts a complex equation into a real-valued one, stripping away the imaginary components that often obscure the truth.

To proceed, we must reach into our mathematical toolkit. We recall three fundamental properties of the modulus: the power rule $|z^n|=|z{|}^n$, the product rule $|z_1{z}_2|=|z_1||z_2|$, and the conjugate rule $|\bar{z}|=|z|$.

These are not just formulas; they are the laws of the complex universe. Applying these, the left side of our equation, $|z^2|$, becomes $|z{|}^2$. On the right side, we split the modulus:

Since $|-i|=1$ and $|\bar{z}|=|z|$, the right side simplifies beautifully to just $|z|$. Our complex equation has now collapsed into the simple, elegant form:

We are almost there, but we must be careful. A common mistake is to simply divide by $|z|$, which would lose a potential solution.

Instead, we bring everything to one side:

Factoring this, we get $|z|(|z|-1)=0$. This gives us two possibilities: $|z|=0$ or $|z|=1$.

This is where the constraint $xy eq 0$ becomes our guide. If $|z|=0$, then $z$ must be the origin, meaning $x=0$ and $y=0$, which makes $xy=0$. This violates our constraint! Thus, we reject $|z|=0$. The only valid solution is $|z|=1$.

We have arrived at the destination. We know $|z|=1$.

The question asks for $|z{|}^2$. Since $|z|=1$, it follows that:

We have navigated the traps, applied the properties, and arrived at the answer with precision. The final result is 1.