Aussie Maths & Science Teachers: Save your time with SmarterEd
It is given that \(\abs{z-1+i}=2\).
What is the maximum possible value of \(\abs{z}\)?
\(\abs{z-1+i}=2 \ \Rightarrow \ \text{circle centre} \ \ (1,-i), \ \ \text {radius}=2\)
\(OA=\sqrt{1^2+1^2}=\sqrt{2}\)
\(AB=2 \ \ \text{(radius)}\)
\(\therefore \abs{z}_{\text{max}}=2+\sqrt{2}\)
\(\Rightarrow C\)
The graph of the circle given by `|z-2-sqrt3i| = 1`, where `z ∈ C`, is shown below.
For points on this circle, find the maximum value of `|z|`. (2 marks)
--- 6 WORK AREA LINES (style=lined) ---
`text(Centre of circle at)\ (2, sqrt3)`
`text(Radius = 1)`
`text(By Pythagoras, line from origin to the centre of the circle)`
`d = sqrt(2^2 + sqrt(3)^2) = sqrt7`
`:. |z|_text(max) = sqrt7 + 1`
The complex number `z` satisfies `| z-1 | = 1.`
What is the greatest distance that `z` can be from the point `i` on the Argand diagram?
`| z-1 | = 1\ \ text{is a circle, centre (1,0), radius 1.}`
`text(Consider the graph:)`
`text(Distance from)\ \(0, i)\ \ text(to the centre)=sqrt2`
`:.\ text(Greatest distance)=sqrt2 + text(radius)=sqrt2+1`
`=> D`