If z = a + bi is a complex number such that |z| = 10, what is the equation relat
Practice Questions
Q1
If z = a + bi is a complex number such that |z| = 10, what is the equation relating a and b?
a^2 + b^2 = 100
a^2 + b^2 = 10
a^2 - b^2 = 100
a^2 + b = 10
Questions & Step-by-Step Solutions
If z = a + bi is a complex number such that |z| = 10, what is the equation relating a and b?
Step 1: Understand that a complex number z can be written as z = a + bi, where a is the real part and b is the imaginary part.
Step 2: The modulus (or absolute value) of the complex number z, denoted as |z|, is calculated using the formula |z| = √(a^2 + b^2).
Step 3: We are given that |z| = 10.
Step 4: Substitute the value of |z| into the modulus formula: √(a^2 + b^2) = 10.
Step 5: To eliminate the square root, square both sides of the equation: (√(a^2 + b^2))^2 = 10^2.
Step 6: This simplifies to a^2 + b^2 = 100.
Step 7: The equation relating a and b is a^2 + b^2 = 100.
Complex Numbers – Understanding the representation of complex numbers in the form z = a + bi, where a is the real part and b is the imaginary part.
Modulus of a Complex Number – The modulus |z| of a complex number is calculated using the formula |z| = √(a^2 + b^2), which relates the real and imaginary parts.
Equations in Complex Analysis – Formulating equations based on the properties of complex numbers, such as their modulus.
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