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# p6 edexcel questions watch

1. 1. A transformation from the z-plane to the w-plane is given by
w=(z-i)/z
show that under this transformation the line Im(z)=1/2 is mapped to the circle with equation |w|=1
Hence, or otherwise, find, in the form w=(az+b)/(cz+d) where a,b,c, and d are C(you know what i mean), the transformation that maps the line Im(z)=1/2 to the circle centre(3-i) and radius 2.

2.
M is the matrix
101
020
431
(a) Find the eigenvalues and corresponding eignvectors for the matrix M.( I've done it)
(b)
A transformation T:R^3->R^3 is represented by the matrix M. Find catrtesian equations of the image of the line with equations
x/2=y=-z/1
under this transformation.
2. (Original post by totaljj)
1. A transformation from the z-plane to the w-plane is given by
w=(z-i)/z
show that under this transformation the line Im(z)=1/2 is mapped to the circle with equation |w|=1
Hence, or otherwise, find, in the form w=(az+b)/(cz+d) where a,b,c, and d are C(you know what i mean), the transformation that maps the line Im(z)=1/2 to the circle centre(3-i) and radius 2.
p + qi
= (x - (1/2)i) / (x + (1/2)i)
= (x - (1/2)i)^2 / [(x + (1/2)i)(x - (1/2)i)]
= (x^2 - 1/4 + xi) / (x^2 + 1/4).

So

p = (x^2 - 1/4) / (x^2 + 1/4),
q = x / (x^2 + 1/4).

So

p^2 + q^2
= [(x^2 - 1/4)^2 + x^2] / (x^2 + 1/4)^2
= (x^2 + 1/4)^2 / (x^2 + 1/4)^2
= 1.

For the second part, you need to simplify 2(z - i)/z + (3 - i).
2.
M is the matrix
101
020
431
(a) Find the eigenvalues and corresponding eignvectors for the matrix M.( I've done it)
(b)
A transformation T:R^3->R^3 is represented by the matrix M. Find catrtesian equations of the image of the line with equations
x/2=y=-z/1
under this transformation.
Choose a point P other than the origin on the line. Find the equation of the line through O and T(P).
3. I dont understand why
For the second part, you need to simplify 2(z - i)/z + (3 - i).
could you explain for me thx
4. (Original post by totaljj)
I dont understand why
For the second part, you need to simplify 2(z - i)/z + (3 - i).
could you explain for me thx
w = (z - i)/z gives you a circle with centre O and radius 1.

w = 2(z - i)/z gives you a circle with centre O and radius 2.

w = 2(z - i)/z + (3 - i) gives you a circle with centre 3 - i and radius 2.

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Updated: June 28, 2004
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