FM101 Locus in the Argand diagrams

Loci in the Argand diagram

WB1 Distance between two points |z−z1|represents the distance between points z and z1

WB2 Loci of a circle with complex numbers

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FM101 Locus in the Argand diagrams

WB3 Given that z satisfies |z−4|=5 a) Sketch the locus of z on an Argand diagramb) Find the values of z that satisfy both |z−4|=5 and Im (z )=0 c) Find the values of z that satisfy both |z−4|=5 and ℜ ( z )=0

WB4 Acomplex number is represented by the point P in the Argand diagram.Given that |z−5−3 i|=3

a) Sketch the locus of Pb) Find the Cartesian equation of this locusc) Find the maximum value of arg z in the interval [−π ,π ]

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FM101 Locus in the Argand diagrams

WB5 Given that the complex number z=x+ yi satisfies |z−12−5i|=3 Find the minimum value of |z| and maximum value of |z|

WB6 Perpendicular bisector

A point is equidistant from 2 points if it lies on the perpendicular bisector of the line segment between them.

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FM101 Locus in the Argand diagrams

WB7 Given that |z−3|=|z+i| a) Sketch the locus of z and find the Cartesian equation of this locusb) Find the least possible value of |z|

WB7 (a) revisited: Alternate method

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FM101 Locus in the Argand diagrams WB8 |z−6|=2|z+6−9 i|

Find the Cartesian equation of the locus of z and sketch it on a Argand diagram

WB9 |z+2+3 i|=|z+1−4 i| Find the Cartesian equation of the locus of z and sketch it on a Argand diagram

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FM101 Locus in the Argand diagrams WB10 Loci using the argument

WB11 sketch the locus of z when arg ( z−(2−i))=π3

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FM101 Locus in the Argand diagrams

WB12 the locus of an arc

WB13 a) sketch the locus of z when arg( z−2 i

z+3 )= π4

b) What would be the equation of the locus of the minor arc?

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FM101 Locus in the Argand diagrams

WB14 generalising / how do we know which side to put the arc?

e.g. sketch the locus of z when arg( z−2 iz+3 )=θ, θ>0

WB15a) sketch the locus of z when arg( z−5

z−1 )=π2

b) Find the centre of the resulting locus

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z

1z

z

1z

2z

FM101 Locus in the Argand diagrams

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FM101 Locus in the Argand diagrams

WB17 You may asked to identify a point satisfying two rules find the complex number z which satisfies both |z−3+2i|=4

and arg ( z−1 )= π4

WB18Max and Min values

The point P represents a complex number z in an Argand diagram. Given that |z+1−i|=1

a) Find a Cartesian equation for the locus of P and sketch it in an Argand diagram b) Find the greatest and least values of |z|

c) Find the greatest and least values of |z−1|

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FM101 Locus in the Argand diagrams

WB19 Sketch the region satisfying |z−4−2i|≤2, |z−4|≤|z−6|,

and arg (z−2−2 i)≤ π4

WB20 a) On the same Argand diagram sketch the loci given by the following equations |z−1|=1

, arg ( z+1 )= π12, arg ( z+1 )=π

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b) Shade on your diagram the region for which |z−1|≤1 and π12≤arg ( z+1 )≤ π

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FM101 Locus in the Argand diagrams

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FM101 Locus in the Argand diagrams

WB21 what is the effect of the transformation m= x2 on the function y=x2

WB22 what is the effect of the transformation w=kz on the locus given by |z|=a Where k is real and positive

WB23 what is the effect of the transformation w=z+z1 on the locus given by |z|=a Where k is real and positive

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FM101 Locus in the Argand diagrams WB24 what is the effect of the transformation w=kz+z1 on the locus

given by |z|=a Where k is real and positive

Summary so far

w = z + z1 where z1 = a + ib represents a translation vector (ab)w = kz represents an enlargement scale factor k centre (0,0)w = kz + z1 where z1 = a + ib represents an enlargement scale factor k centre (0,0),

followed by a translation vector (ab)

Other transformations are less obvious to spot, but can be interpreted using the same method as before

• Make z the subject• Substitute into locus expression & rearrange to a familiar locus

Typically, this requires introducing w=u+iv and some manipulation

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FM101 Locus in the Argand diagrams

WB25 a) Show that the transformation w= z−1z maps |z−1|=1 in the

z-plane onto |w|=|w−1| in the w-planeThe region |z−1|≤1 in the z-plane is mapped onto the region T in the w-planeb) Shade the region T on an argand diagram

WB26 The transformation T from the z-plane to the w-plane is given by w=3 z−2z+1

Show that the image, under T, of the circle with equation x2+ y2=4 in the z-plane is a circle C in the w-plane. State the centre and radius of C.

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FM101 Locus in the Argand diagrams

WB27 A transformation T of the z-plane to the w-plane is given by w= iz−21−z

Show that as z lies on the real axis in the z-plane, then w lies on the line L in the w-plane. Sketch L on an Argand diagram

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FM101 Locus in the Argand diagrams WB28 The transformation T from the z-plane to the w-plane is given by w= z+ i

z

a) The transformation T maps the points on the line with equation y=x in the z-plane to points on the line L in the w-plane. Find an equation of L

b) Show that the image, under T, of the line with equation x+ y+1=0 in the z-plane is a circle C in the w-plane. State the centre and radius of C.

c) Sketch l and C on the same Argand diagram

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FM101 Locus in the Argand diagrams

WB29 The transformation T from the z-plane to the w-plane is defined

by w=2(1+i)z+2

Show that the locus of P in the z-plane is mapped to part of a straight line

in the w-plane, and show this in an Argand diagram

WB30Evil exam Q

The point Prepresents the complex number zon an Argand diagram, where the locus of P as z varies is the circle C with Cartesian equation x2+( y−1)2=4The locus of Pas zvaries is the curve C.

A transformation Tfrom the z-plane to the w-plane is given by w= z+i3+iz

, z ≠3 i

The point Qis mapped by Tonto the point R. Given that Rlies on the real axis,(c) show that Qlies on C.

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FM101 Locus in the Argand diagrams

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