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Question
If z is a complex number, then ______.
Options
|z2| > |z|2
|z2| = |z|2
|z2| < |z|2
|z2| ≥ |z|2
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Solution
If z is a complex number, then |z2| = |z|2.
Explanation:
Let z = x + yi
|z| = |z + yi| and |z|2 = |x + yi|2
⇒ |z|2 = x2 + y2 ......(i)
Now z2 = x2 + y2i2 + 2xyi
z2 = x2 – y2 + 2xyi
|z|2 = `sqrt((x^2 - y^2)^2 + (xy)^2)`
= `sqrt(x^4 + y^4 - 2x^2 y^2 + 4x^2 y^2)`
= `sqrt(x^4 + y^4 + 2x^2 y^2)`
= `sqrt((x^2 + y^2)^2`
So |z2| = x2 + y2 = |z|2
So |z2| = |z|2
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