- #1

CharlesL

- 17

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## Homework Statement

Question(1) : Find the Cartesian equation of Re[ z - i / z + 1 ] = 0. If the locus is a circle, give its radius and the coordinates of its center.

## The Attempt at a Solution

Workings : So I attempted to solve the problem and my workings are as below

... Since Re = Real part,

Let z = x + iy

Re[ x + iy - i / x + iy + 1 ] = 0

x/x+1 = 0

x = 0

Right here I am assuming that the locus is at all the points of the line x=0.

NEXT, to obtain the radius and the coordinates of the center of the locus(circle),

[modulus] z - i/ z + 1 [modulus] = 0

[modulus] z - i [modulus] = 0

[modulus] x + iy - i [modulus]= 0

square root[ (x - 0)

^{2}+ (y - 1)

^{2}] = 0

(x - 0)

^{2}+ (y - 1)

^{2}= 0

Therefore, the radius of the circle is 0 and the coordinates of the center is ( 0, 1 )

## Homework Statement

Question (2) : Obtain all the real solutions of the following equation: 9 sinh 4x - 82 sinh 3x + 9 sinh 2x = 0 . Show all your derivations.

## The Attempt at a Solution

I first subsituted [ e

^{x}- e

^{-x}/ 2 ] into all the sinh available in the equation with their specific value of x.

9[ e

^{4x}- e

^{-4x}/ 2 ] - 82[ e

^{3x}- e

^{-3x}/ 2 ] + 9[e

^{2x}- e

^{-2x}/ 2 ] = 0

I multiply the whole equation by 2 and decided to multiply the integer outside of each boxes,

9e

^{4x}- 9e

^{-4x}- 82e

^{3x}+ 82e

^{-3x}+ 9e

^{2x}- 9e

^{-2x}= 0

Then I tried to separate e

^{4x}to e

^{4}. e

^{x}and regroup the ones with e

^{x}and e

^{-x}

[ 9e

^{4}- 82e

^{3}+ 9e

^{2}] e

^{x}= [ 9e

^{4}- 82e

^{3}+ 9e

^{2}] e

^{-x}

Then I multiplied both sides with e

^{x}

[e

^{x}]

^{2}= 1

e

^{x}= square root + of 1 (chosed only +ve value as the question mentioned about real solutions)

Then I applied ln

x ln e = ln square root of 1

x = 0

Any comments would be a great help and much appreciated. Thank you in advance and have a nice day.

Regards

Charles

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