Animated Solution for Mathematics - Complex Numbers: If $a>0$ and $z=\frac{(1+i{)}^2}{a-i}$, has magnitude $\sqrt{2/5}$, then $z$ is equal to :

Welcome, fellow traveler on the path to JEE mastery! Today, we are going to unravel a problem that might look like a tangled mess of fractions and imaginary units, but beneath the surface, it is a beautiful exercise in geometric intuition and algebraic elegance.

We are given a complex number $z=\frac{(1+i{)}^2}{a-i}$ with the condition that $a>0$ and its magnitude $|z|=\sqrt{\frac{2}{5}}$. Our mission is to find the standard form of $z$. Let's begin.

Simplifying the Numerator

Before we even think about the magnitude, let's look at that numerator: $(1+i{)}^2$. It looks like a classic expansion waiting to happen.

Using the identity $(A+B{)}^2=A^2+B^2+2AB$, we expand this as $1^2+i^2+2(1)(i)$. Now, recall the fundamental definition of the imaginary unit: $i^2=-1$.

Substituting this in, we get $1-1+2i$. The $1$ and $-1$ cancel out, leaving us with a remarkably simple numerator: $2i$. Our complex number $z$ has now transformed into:

The Power of the Modulus Property

Now, we need to handle the magnitude $|z|=\sqrt{\frac{2}{5}}$. Instead of performing complex division, we use the powerful modulus property:

This property is a lifesaver! It allows us to calculate the modulus of the numerator and denominator separately.

The modulus of the numerator $|2i|$ is simply $\sqrt{0^2+2^2}=2$. The modulus of the denominator $|a-i|$ is $\sqrt{a^2+(-1{)}^2}=\sqrt{a^2+1}$. Thus, we have:

Solving for $a$

We equate our expression to the given magnitude:

To clear the radicals, we square both sides:

Cross-multiplying gives us $20=2(a^2+1)$, which simplifies to $20=2a^2+2$. Subtracting $2$ from both sides, we get $18=2a^2$, or $a^2=9$.

Since the problem explicitly states $a>0$, we must reject $a=-3$ and accept $a=3$. This is a crucial step—never ignore the constraints!

The Final Transformation

With $a=3$, our complex number becomes $z=\frac{2i}{3-i}$. To write this in the standard form $x+iy$, we rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is $3+i$.

This yields:

The numerator becomes $6i+2i^2=-2+6i$. The denominator becomes $3^2-i^2=9-(-1)=10$.

Finally, we split the fraction:

This simplifies to our final answer:

Look at what we have achieved. We took a daunting expression and, through systematic simplification and the application of fundamental properties, reduced it to a clean, standard complex number.

The key takeaway here is to always look for properties—like the modulus property of division—that can simplify your path before you dive into heavy algebra. Keep practicing, stay curious, and remember: every complex problem is just a series of simple steps waiting to be taken.