Moderate -0.5 This is a straightforward locus question requiring substitution of z = x + iy into the modulus equality, expanding both sides, and simplifying to get a Cartesian equation (which will be a perpendicular bisector). The geometric interpretation is direct once the equation is found. Standard Further Maths technique with minimal problem-solving required.
5. (a) The complex number \(z\) is represented by the point \(P ( x , y )\) in an Argand diagram. Given that
$$| z - 3 + 2 i | = | z - 3 |$$
find the equation of the locus of \(P\).
(b) Give a geometric interpretation of the locus of \(P\).
It is the perpendicular bisector of the line joining the points \((3, -2)\) and \((3, 0)\)
B1
OR: The locus of P is all the points which are equidistant from \((3, -2)\) and \((3, 0)\)
(B1)
## Question 5:
### Part a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $|x + iy - 3 + 2i| = |x + iy - 3|$ leading to $|(x-3) + i(y+2)| = |(x-3) + iy|$ | M1 | |
| $\sqrt{(x-3)^2 + (y+2)^2} = \sqrt{(x-3)^2 + y^2}$ | m1 | oe |
| $x^2 - 6x + 9 + y^2 + 4y + 4 = x^2 - 6x + 9 + y^2$, giving $4y + 4 = 0$ | A1 | Mark final answer; sight of answer only M1m1A1 |
| $y = -1$ | | |
### Part b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| It is the perpendicular bisector of the line joining the points $(3, -2)$ and $(3, 0)$ | B1 | |
| OR: The locus of P is all the points which are equidistant from $(3, -2)$ and $(3, 0)$ | (B1) | |
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5. (a) The complex number $z$ is represented by the point $P ( x , y )$ in an Argand diagram. Given that
$$| z - 3 + 2 i | = | z - 3 |$$
find the equation of the locus of $P$.\\
(b) Give a geometric interpretation of the locus of $P$.\\
\hfill \mbox{\textit{WJEC Further Unit 1 2022 Q5 [4]}}