Question

Find the invariant points of the transformation w = 2z+6/z+7
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Answer 1

Answer:

Invariant points of a bilinear transformation w=1+iz1−iz is

⇒1+iz=z(1−iz)

⇒1+iz=z−iz2.

⇒iz2+iz−z+1=0.

iz2+z(i−1)+1=0.

Answer 2

The invariant points of the transformation w are - 6 & 1

Given :

The transformation w = 2z + 6/z + 7

To find :

The invariant points of the transformation w

Solution :

Step 1 of 2 :

Write down the given transformation

Here the given transformation is

$\displaystyle \sf{ w = \frac{2z + 6}{z + 7} }$

Step 2 of 2 :

Find the invariant points of the transformation

For the invariant points of the transformation w we have

$\displaystyle \sf{ w = z }$

$\displaystyle \sf{ \implies \frac{2z + 6}{z + 7} = z}$

$\displaystyle \sf{ \implies {z}^{2} + 7z = 2z + 6 }$

$\displaystyle \sf{ \implies {z}^{2} + 5z - 6 = 0 }$

$\displaystyle \sf{ \implies {z}^{2} + (6 - 1)z - 6 = 0 }$

$\displaystyle \sf{ \implies {z}^{2} + 6z -z - 6 = 0 }$

$\displaystyle \sf{ \implies z(z + 6) - 1(z + 6) = 0 }$

$\displaystyle \sf{ \implies (z + 6) (z - 1) = 0 }$

$\displaystyle \sf{ \implies z = - 6 \:, \: 1}$

Hence the invariant points of the transformation w are - 6 & 1

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