smc-7431-20-Perp Bisector

Complex Numbers, EXT2 N2 2025 HSC 12c

Sketch the region of the complex plane defined by \(\abs{z+5-i}>\abs{z-3+3 i}\). (3 marks)

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\(\abs{z+5-i}=\abs{z-(-5+i)}\)

\(\abs{z-3+3 i}=\abs{z-(3-3 i)}\)

\(\text{Find equation of} \ \perp \ \text{bisector between}\ \ (-5,1) \ \ \text{and}\ \ (3,-3):\)

\(\text{Midpoint} \equiv \left(\dfrac{-5+3}{2}, \dfrac{1-3}{2}\right) \equiv (-1,-1)\)

\(m=\dfrac{-3-1}{3+5}=-\dfrac{1}{2} \ \Rightarrow \ m_{\perp}=2\)

\(\text{Equation of line} \ \ m=2 \ \ \text{through}\ \ (-1,-1):\)

\(y+1\)	\(=2(x+1)\)
\(y\)	\(=2 x+1\)

\(\text{Sketch:}\ \abs{z+5-i}>\abs{z-3+3 i}\)

Two complex numbers, `u` and `v`, are defined as `u = −2-i` and `v = −4-3i`.

Express the relation `|z-u| = |z-v|` in the cartesian form `y = mx + c`, where `m, c ∈ R`. (3 marks)

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Plot the points that represent `u` and `v` and the relation `|z-u| = |z-v|` on the Argand diagram below. (2 marks)
State a geometrical interpretation of the graph of `|z-u| = |z-v|` in relation to the points that represent `u` and `v`. (1 mark)

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Sketch the ray given by `text(Arg)(z-u) = pi/4` on the Argand diagram in part b. (1 mark)

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In Cartesian form, write down the function that describes the ray `text(Arg)(z-u) = pi/4`. (1 mark)

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a. `y = −x-5`

c. `|z-u| = |z-v|\ text(is the graph of the perpendicular bisector of the)`

`text(line joining)\ u and v.`

e. `f: (−2, ∞) ->, f(x) = x + 1`

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a. `text(Let)\ \ z = x + iy`

`z-u = x + 2 + iy + i`

`z-v = x + 4 + iy + 3i`

`|z-u| = |z-v|`

c. `|z-u| = |z-v|\ text(is the graph of the perpendicular bisector of the)`

`text(line joining)\ u and v.`

e. `text(Arg)(z-u) = pi/4 =>\ text(gradient) = 1, ytext(-intercept at)\ (0, 1)`

`:. f: (−2, ∞) -> RR, \ f(x) = x + 1`

Sketch the region in the Argand diagram where `|\ z\ | ≤ |\ z-2\ |` and `−pi/4 ≤ text(arg)\ z ≤ pi/4`. (3 marks)

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`text(See Worked Solutions.)`

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`text(Let)\ \ z=x+iy`

`−pi/4 ≤ text(arg)\ z ≤ pi/4`