Complex number - geometrical representation (1 Viewer)

[YBK]

Question 15 from cambridge.

i can do the first part, but stuck on the second... the question is as follows:

a) obtain in the form a + ib the roots of the equation x^2 + 2x + 3 = 0
Find the modulus and argument of each root and represent the roots on an Argand diagram by the points A and B

b) Let H and K be the points representing the roots of x^2 + 2px + q = 0 where p and q are real and p^2 < q. Find the algebraic relation satisfied by p and q in each of the following cases:

i) angle HOK is a right angle
ii) A, B, H, and K are equidistant from O.


Solution to first part:

x^2 + 2x + 3 = 0

x= -1 +- root(2) i

-1 + root(2) i
Modulus = root3
arg z = pie - tan^-1 root2

-1 - root(2) i
Modulus = root 3
arg z = - tan^-1 root2



Thanks!!!

 

[pLuvia]

b)

x² + 2px + q
Using quad formula
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You will eventually get this

x = -p+ iroot(q-p²)

.:
H(-p, root(q-p²))
Q(-p, -root(q-p²))

Hence
-p = root(q-p²
p² = q - p²
2p² = q

[ii] Using your drawn diagram from part (a)

OA = root(1+2) = root 3

OB = root 3
OH² = p² + q - p² = q

From x²+2x+3=0
q = 3
p = 1

 

[YBK]

b)

x² + 2px + q
Using quad formula
|
|
|
|

You will eventually get this

x = -p+ iroot(q-p²)

.:
H(-p, root(q-p²))
Q(-p, -root(q-p²))

Hence
-p = root(q-p²
p² = q - p²
2p² = q

[ii] Using your drawn diagram from part (a)

OA = root(1+2) = root 3

OB = root 3
OH² = p² + q - p² = q

From x²+2x+3=0
q = 3
p = 1

awsome, thank you!
I'll have to print that out and try to undertand it

 

[pLuvia]

lol, no probs