Problem
Let z be a complex number with non-zero imaginary part. If (2 + 3z + 4z²)/(2 − 3z + 4z²) is a real number, then find the value of |z|².
Solution
Let’s look at four different methods to arrive at the answer. Before that, we’ll simplify the given expression.
(2 + 3z + 4z²)/(2 − 3z + 4z²)
= (2 – 3z + 4z² + 6z)/(2 − 3z + 4z²)
= 1 + 6z/(2 − 3z + 4z²)
= 1 + 6/(2/z − 3 + 4z)
How did that help?
Since 1 + 6/(2/z − 3 + 4z) is real, 6/(2/z − 3 + 4z) is also real.
Therefore, 2/z − 3 + 4z must also be real, which means that 2/z + 4z must be real, and finally, 1/z + 2z must be real.
And, we need to deal with only this expression.
Method 1
For a complex number to be real, it must be equal to its conjugate.
That means 1/z + 2z =1/\(\bar{z}\) + 2\(\bar{z}\).
On rearranging, we’ll get 1/z – 1/\(\bar{z}\) = 2(\(\bar{z}\) – z).
This means (\(\bar{z}\) – z)/z\(\bar{z}\) = 2(\(\bar{z}\) – z).
Now, since z has a non-zero imaginary part, it means that z – \(\bar{z}\) is non-zero, and we can cancel out that terms from both sides.
We’ll get z\(\bar{z}\) = 1/2, which means |z|2 = 1/2.
Method 2
Since 1/z + 2z is real, let’s equate it to a real number k, to get 2z + 1/z = k.
This results in a quadratic, 2z2 – kz + 1 = 0, which gives z = [k ± √(k2 – 8)]/4.
Now, since z is a complex number, and k is real, k2 – 8 must be negative and the two value of z must be complex conjugates.
So, z\(\bar{z}\) = [k + √(k2 – 8)]/4 x [k – √(k2 – 8)]/4, which equals [k2 – (k2 – 8)]/16 or 1/2.
Method 3
Let’s assume z to be x + iy, where y is non-zero.
Then, the previous expression becomes 1/(x + iy) + 2(x + iy), or (x – iy)/(x2 + y2) + 2(x + iy).
Since this is real, its imaginary part must equal 0. That is, -y/(x2 + y2) + 2y = 0.
This gives us 2y(x2 + y2) = y. On cancelling out the non-zero term y, we get (x2 + y2) = 1/2.
In other words, |z|2 = 1/2.
Method 4
Let’s assume z = |z|eiθ. Then, our expression becomes e-iθ/|z| + 2|z|eiθ.
Using eiθ = cosθ + isinθ, the previous expression changes to (2|z| + 1/|z|)cosθ + i(2|z| – 1/|z|)sinθ.
Since this expression is real, it’s imaginary part must be zero.
Or, (2|z| – 1/|z|)sinθ = 0. This gives 2|z| = 1/|z|, and once again, |z|2 = 1/2. (Why can’t sinθ be zero?)
Comments
A simple problem. But things would’ve gotten complicated if we hadn’t done the simplification at the beginning. Coming to the methods, I’d suggest the first one, which is the quickest. Still, good to know the other three.
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