What loci is represented by the following equation

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In summary, the equation |z+1| = |z-1| represents two circles with radius 1 centered at (1,0) and (-1,0) on the complex plane. The points of intersection between these circles are the solutions to the equation and can be any point along the y-axis. The loci origins can be any combination of a = b, and crossing circles is a valid method to find solutions.
  • #1
thomas49th
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What loci is represented by the following equation:
|z+1| = |z-1|

I believe I can get the answer, I am just slightly confused by what is going on?

I turned this to

|z+1| / |z-1| = 1, which means |z+1| = 1 and |z-1| = 1. Doe this mean we can draw 2 circles of radius 1 at the points (1,0) and (-1,0)? The only place they touch is the origin? Is this correct?

OR
Letting z = x + iy

[tex]\frac{\sqrt{(x+1)^{2} + y^{2}}}{\sqrt{(x-1)^{2}+y^{2}}} = 1[/tex]

[tex]\frac{(x+1)^{2} + y^{2}}{(x-1)^{2}+y^{2}} = 1[/tex]

[tex]4x = 0[/tex]

x = 0
That implies y can be anypoint along the y-axis (at x= 0). Is this correct?

Finally are the actual locations of the loci origins actually -1 and 1, because surely z doesn't have to be x + iy, I could be 5x + 6iy

Thanks
Thomas
 
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  • #2
That implies y can be anypoint along the y-axis (at x= 0). Is this correct?

yes.

Finally are the actual locations of the loci origins actually -1 and 1, because surely z doesn't have to be x + iy, I could be 5x + 6iy
?
 
  • #3
Quinzio said:
yes.


?

Okay, forget the last bit, but why is my method of drawing circles and finding where they cross not valid?
 
  • #4
thomas49th said:
Okay, forget the last bit, but why is my method of drawing circles and finding where they cross not valid?

You're given

a= b

You wrote
a/b = 1 which is correct.

Then you wrote
a=1, b=1
why ?
a=3, b=3 is valid, too, any combination of a = b is valid.
It's ok to cross circles, but their radious can be any number.
 
  • #5
ahhhhhhhhhhhh I think I seeeeee
Cheers
 

Related to What loci is represented by the following equation

What loci is represented by the following equation?

The following equation represents a mathematical relationship between variables, and does not specifically refer to a specific loci. Loci, or loci points, are specific locations on a coordinate plane that represent the solutions to an equation. Therefore, the equation itself does not represent a specific loci.

How do you find the loci represented by an equation?

To find the loci represented by an equation, you can graph the equation on a coordinate plane and identify the points where the equation intersects with the x and y axes. These points will represent the loci for the equation.

Can an equation have multiple loci?

Yes, an equation can have multiple loci. This means that there can be multiple points on a coordinate plane that satisfy the equation. These points may form a line, a curve, or even a complex shape.

What type of equations are commonly associated with loci?

Equations that involve variables raised to the first or second power, such as linear and quadratic equations, are commonly associated with loci. This is because these types of equations can be easily graphed on a coordinate plane to identify the loci points.

What is the significance of identifying loci in an equation?

Identifying loci in an equation can help us understand the relationship between variables and the solutions to the equation. It can also help us visualize the solutions and how they may change as the variables change. This can be useful in various scientific fields such as physics, chemistry, and biology.

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