foundations of 4U maths book - S.K.Patel (1 Viewer)

[*Pooja*]

hey ppl. does anyone have the solutions to exercise 4O in this textbook. its the revision exercise for complex numbers and some of them are way too hard for me.

the stupid topic happens to be tested in my half yrlies.

pm me if u have the answers pls. thanks.

 

[nike33]

post the questions up (dont have the book) and ill give em a go (as im sure will other people )

 

[aud]

Originally posted by *Pooja*
hey ppl. does anyone have the solutions to exercise 4O in this textbook. its the revision exercise for complex numbers and some of them are way too hard for me.

the stupid topic happens to be tested in my half yrlies.

pm me if u have the answers pls. thanks.

I have Maths Extension 2 2nd Edition by Patel... think it's Foundations republished, cause 4O is still the revision for Complex... and all the answers are there on pages 357 - 358...

If the answers to
3. a. 3 - 2i
3. b. locus is either a point (3, -2) or the circle (x - 3) + (y + 2) = 1
4. a. x = 3/2, y = 2 or x = -3/2, y = -2
4. b. i. x + 2y + 2 = 0
4. b. ii. circle x + y + 2x + y = 0, C (-1, -1/2), r = rt5/2
make sense, let me know

 

[*Pooja*]

questions: 1e (i dont know where i'm going wrong), 2, 3b

thanks.

 

[Azn_Phoenix]

Hey does ne1 have worked solutions 4 Patel?? I really need them...plz

 

[redslert]

yes i did the exercies in patel...
on paper, not about to scan it
not about to give it out

would be better if u had problems, post them here and people will get back to you

 

[*Pooja*]

how do u do this one (its question 2 in this book):

the points z1, z2 and z3 are three complex numbers that lie on a circle passing through the origin. Prove that the points which represent 1/z1, 1/z2 and 1/z3 are collinear.

 

[CM_Tutor]

The points are collinear if the line through 1 / z₁ and 1 / z₂ is parallel to the line through 1 / z₁ and 1 / z₃. Thus, we need to show that arg[(1 / z₁) - (1 / z₂)] = arg[(1 / z₁) - (1 / z₃)]

Start by taking this statement, and showing it is the same as arg(z₂ - z₁) - arg z₂ = arg(z₃ - z₁) - arg z₃.

Now, draw a diagram of a circle, passing through O and z₁ and z₂ and z₃, and use it to prove (geometrically) that the above statement is true.

Edit: Note - the above proof requires the points O, z₁, z₂ and z₃ to be placed in this order around the circle. This may be done without loss of generality. Alternately, if not placed in this order, then the proof needs to be extended to state that either arg[(1 / z₁) - (1 / z₂)] = arg[(1 / z₁) - (1 / z₃)] or
arg[(1 / z₁) - (1 / z₂)] = pi - arg[(1 / z₁) - (1 / z₃)]. Depending on the relative positions of O, z₁, z₂ and z₃, exactly one of these statements is true (provable geometrically), and either will result in 1 / z₁, 1 / z₂ and 1 / z₃ being collinear.

Further note - can you construct a geometric proof that the line on which 1 / z₁, 1 / z₂ and 1 / z₃ lie does not pass through the origin?

 

[*Pooja*]

thanks! i'll try understand whats done here. (im a slow learner)

 

[ngai]

ohh, that question..
consider a moving pt P(z) on the circle, and the fixed pt Q(w), where OQ is the diameter
then u have the points A(1/z) which is variable and B(1/w) which is fixed
using geometry and modulus stuffs, prove OPQ similar to OAB
then using angle in semicircle, angle OPQ = angle OBA = 90 for all z
so for all z on the circle, 1/z lies on the line through B perpendicular to OB
and so its true for z=z1, z2, z3

 

[Pianpupodoel]

Sorry to raise an *extremely* old topic from the dead, i just did a search thingy and found this.

About question two in 4O, what about this example -

Taking the unit circle, z1 = i, z2 = 1, z3 = -i

1/z1 = -i, 1/z2 = 1, 1/z3 = i.

It would seem quite obvious they do not lie on the same line. Hence the question is wrong?

 

[icycloud]

It would seem quite obvious they do not lie on the same line. Hence the question is wrong?

Your circle doesn't pass through the origin (as the question stipulates).

In fact it can be proved that if the circle z doesn't pass through the origin, then 1/z is the locus of another circle (there's another thread about that somewhere here).

So basically: if z is a circle which passes through the origin, then 1/z is a line.
if z is a circle which does not pass through the origin, then 1/z is a circle.

 

[Pianpupodoel]

Oh. Well isnt that stupid of me.

Thank you.

 

[shsshs]

richard n aaron. such rivals yet such friends

 

[davidyin92]

what is the a and b of the hyperbola x^2 and 4y^2 = 1
I can't seem to get the right answer..its in exercise 6B question 3

 

[independantz]

what is the a and b of the hyperbola x^2 and 4y^2 = 1
I can't seem to get the right answer..its in exercise 6B question 3

lol, you reopened a 2 year old thread 0_0

i'd think a=1, b=(1/2)

since (y/(1/2))^2=4y^2

 

[davidyin92]

lol, thanks for the help i got it now
I'll make sure not to revive a year old thread again haha