Some questions about complex no and polynomial (1 Viewer)

[schmeichung]

1.
If one root of x^3+mx^2+Nx+p=0 is the sum of the other roots show that m^3-4mN+8p=0

For this one I let A,B,A+B are the roots
and I get those equations:
2(A+B)=-m ------>[1]
A(A+B)+B(A+B)=N --> (A+B)^2=N ------>[2]
AB(A+B)=-p ------->[3]
then [1]^3, [2]*(4)[1], 8*[3]
I add all them up
and get -8(A+B)^3 + 8(A+B)^3 + -8(AB(A+B)) = m^3-4mN+8p
then I dont know how to proceed further..

2.
Let z1=3+6i and z2=-3-6i
Show that the locus specified by |z-z1|=2|z-z2| is a circle.
Give the co-ordinates of the centre and the length of is radius.

For this one I tried let z=x+iy
then I expand the modulus out and get:
(x-3)^2 + (y-6)^2 = 4 [ (x+3)^2 + (y+6)^2 ]
and end up get (x+5)^2+(y+10)^2=80
I am not sure if this is correct (I have no confident on this.. ) becoz the radius is pretty weird (not an integer)

3.
A(z1), B(z2) and C(z3) are three points on the complex plane.
Prove that:
angle(ABC) = arg(z1-z2) - arg(z3-z2)

I have no idea how to do this one..


Thank you all for helping me

 

[schmeichung]

I got the 1st one now
I missed AB in the 2nd line, sum of products of 2 roots

1.
If one root of x^3+mx^2+Nx+p=0 is the sum of the other roots show that m^3-4mN+8p=0

For this one I let A,B,A+B are the roots
and I get those equations:
2(A+B)=-m ------>[1]
A(A+B)+B(A+B)=N --> (A+B)^2=N ------>[2]
AB(A+B)=-p ------->[3]
then [1]^3, [2]*(4)[1], 8*[3]
I add all them up
and get -8(A+B)^3 + 8(A+B)^3 + -8(AB(A+B)) = m^3-4mN+8p
then I dont know how to proceed further..

2.
Let z1=3+6i and z2=-3-6i
Show that the locus specified by |z-z1|=2|z-z2| is a circle.
Give the co-ordinates of the centre and the length of is radius.

For this one I tried let z=x+iy
then I expand the modulus out and get:
(x-3)^2 + (y-6)^2 = 4 [ (x+3)^2 + (y+6)^2 ]
and end up get (x+5)^2+(y+10)^2=80
I am not sure if this is correct (I have no confident on this.. ) becoz the radius is pretty weird (not an integer)

3.
A(z1), B(z2) and C(z3) are three points on the complex plane.
Prove that:
angle(ABC) = arg(z1-z2) - arg(z3-z2)

I have no idea how to do this one..


Thank you all for helping me

 

[wogboy]

1.

A(A+B)+B(A+B)=N --> (A+B)^2=N ------>[2]

Try A(A+B) + B(A+B) + AB = N.

3.

A(z1), B(z2) and C(z3) are three points on the complex plane.
Prove that:
angle(ABC) = arg(z1-z2) - arg(z3-z2)

angle ABC is the angle between the intervals CB and BA. CB = z3 - z2, BA = z1 - z2. The angle BA makes with the positive x-axis is Arg(z1 - z2) and the angle CB makes with the positive x-axis is Arg(z3 - z2). So the angle between CB and BA is Arg(z3 - z2) - Arg(z1 - z2).

 

[wogboy]

2.

Continuing from (x-3)^2 + (y-6)^2 = 4*{(x+3)^2 + (y+6)^2}
x^2 - 6x + 9 + y^2 - 12y + 36 = 4*{x^2 + 6x + 9 + y^2 + 12y + 36}
3x^2 + 30x + 3y^2 + 60y + 135 = 0
x^2 + 10x + y^2 + 20y + 45 = 0
(x + 5)^2 - 25 + (y + 10)^2 - 100 + 45 = 0
(x + 5)^2 + (y + 10)^2 = 80

Looks like you're correct.

 

[schmeichung]

Oh got it!
Thank you very much!

 

[ngai]

radius isn't that weird...at least its the square root of an integer..
u can't expect all questions to give u nice numbers like 3,4,5