This Hint is for students who have only recently met complex
numbers.
What can you say about the product of the roots of the quadratic
equation x2 + px + q = 0?
If the roots of this quadratic equation, where p and q are real
coefficients, are the complex numbers z1 and z2 explain why
z1 and z2 are complex conjugates.
Explain why the square of the distance from the origin of the point
u+iv in the Argand diagram is given by the product of complex
conjugates (u+iv)(u-iv).
What does this tell you about the locus of the complex roots of x2 +px +q = 0 as you change p keeping q constant?
Was your conjecture correct?