Important question

Dec 2019
18
0
Abeokuta
Let P represent the complex number z, Q represent the complex number z + 1/z. Show that if P moves on the circle then |z| = 2, Q moves on the ellipse then,
x^2/25 + y^2/9 = 1/4
 

skipjack

Forum Staff
Dec 2006
21,321
2,390
If z = 2cos(θ) + 2sin(θ)i, z + 1/z = (5/2)cos(θ) + (3/2)sin(θ)i,
so Q is on the ellipse with equation x²/25 + y²/9 = 1/4.
 
Dec 2019
18
0
Abeokuta
If z = 2cos(θ) + 2sin(θ)i, z + 1/z = (5/2)cos(θ) + (3/2)sin(θ)i,
so Q is on the ellipse with equation x²/25 + y²/9 = 1/4.
I don't understand ooooooouuuu please let it be explanatory
 

skipjack

Forum Staff
Dec 2006
21,321
2,390
Do you understand what |z| means?
 

skipjack

Forum Staff
Dec 2006
21,321
2,390
If z has real part x and imaginary part y, z = |z|(cos(θ) + sin(θ)i), where $-\pi\small <$ θ ${\small\leqslant}\, \pi$,
and (unless z = 0) 1/z = (1/|z|)(cos(θ) - sin(θ)i).

It's easy to verify that the product of those expressions is 1 (but you can assume their validity anyway).

The amount of detail that you give in your answer depends on what you've already learnt, but you should obtain the real and imaginary parts of z + 1/z by using the equations I've given above.

Thus x and y can be thought of as coordinates for a point on an x-y plane (known as an Argand diagram). You can assume that |z| = 2 means the point P(x, y) lies on a circle of radius 2 and equation x² + y² = 2².

Let Q be a point on the same Argand diagram representing the complex number z + 1/z.
You can verify algebraically that the corresponding equation for the locus of Q is x²/25 + y²/9 = (1/2)² = 1/4, where x and y are now the real and imaginary parts of z + 1/z.

You can assume that is the equation of an ellipse.
 
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