- Show that the solutions of `2z^2 + 4z + 5 = 0`, where `z ∈ C`, are `z = −1 ± sqrt6/2 i`. (1 mark)
- Plot the solutions of `2z^2 + 4z + 5 = 0` on the Argand diagram below. (1 mark)
Let `|z + m| = n`, where `m, n ∈ R`, represent the circle of minimum radius that passes through the solutions of `2z^2 + 4z + 5 = 0`.
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- Find `m` and `n`. (2 marks)
- Find the cartesian equation of the circle `|z + m| = n`. (1 mark)
- Sketch the circle on the Argand diagram in part a.ii. Intercepts with the coordinate axes do not need to be calculated or labelled. (1 mark)
- Find all values of `d`, where `d ∈ R`, for which the solutions of `2z^2 + 4z + d = 0` satisfy the relation `|z + m| <= n`. (2 marks)
- All complex solutions of `az^2 + bz + c = 0` have non-zero real and imaginary parts.
Let `|z + p| = q` represent the circle of minimum radius in the complex plane that passes through these solutions, where `a, b, c, p, q ∈ R`.
Find `p` and `q` in terms of `a, b` and `c`. (2 marks)