If $z=x+iy,xy\ne <!--\; \ne \; -->0$, satisifes the equation $z^2+i\stackrel{¯<!--\; ¯\; -->}{z}=0$, then $\left|z^2\right|$ is equal to :

The correct option is 1.

Let us analyze the given equation step-by-step. We are given a complex number $z=x+iy$ where $xy\ne 0$ (meaning both the real and imaginary parts of $z$ are non-zero), satisfying the equation:

$z^2+i\overline{z}=0$

We want to find the value of $\left|z^2\right|$, which is equivalent to ${\left|z\right|}^2$.

First, we rearrange the given equation by moving the term involving the conjugate of $z$ to the right-hand side:

$z^2=-i\overline{z}$

Next, we take the absolute value (modulus) of both sides of this equation:

$\left|z^2\right|=|-i\overline{z}|$

Using the properties of the modulus, we can simplify both sides. Specifically, the modulus of a product is the product of the moduli, and the modulus of the conjugate of a complex number is equal to the modulus of the number itself:

$|-i\overline{z}|=|-i|·\left|\overline{z}\right|$

Since $|-i|=1$ and $\left|\overline{z}\right|=\left|z\right|$, and also $\left|z^2\right|={\left|z\right|}^2$, our equation simplifies to:

${\left|z\right|}^2=\left|z\right|$

This equation can be rewritten as:

${\left|z\right|}^2-\left|z\right|=0$

$\left|z\right|\left(\right|z|-1)=0$

This yields two possible values for the modulus of $z$:

$\left|z\right|=0$ or $\left|z\right|=1$

We are given that $xy\ne 0$, which means $z$ cannot be zero. Therefore, $\left|z\right|\ne 0$. This leaves us with the only valid solution:

$\left|z\right|=1$

Finally, we calculate $\left|z^2\right|$:

$\left|z^2\right|={\left|z\right|}^2=1^2=1$