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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
2
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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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