Find the locus of points that satisfy
(a) |z+1| = |z+i|
(b) Re z = |z-1|
and sketch them in the complex plane.
since |z-c|=r is a circle of radius r,
|z+1| = |z+i| is the intersection of two circles.
If z = x+yi,
Re z = x
|z-1|^2 = (x-1)^2 + y^2
So, you need x^2 = (x-1)^2
correction on #1. What I gave was the solution for a particular radius. Since we have to consider all possible such circles, the locus is the perpendicular bisector of the line joining centers of the circles. All points on that line are equidistant from the two centers.
To find the locus of points that satisfy the given equations, we need to manipulate the expressions to isolate the variable z and then interpret the results geometrically.
(a) |z+1| = |z+i|
Let's start by squaring both sides of the equation:
|z+1|^2 = |z+i|^2
Using the properties of complex numbers, we know that |z|^2 = z * conjugate(z), where conjugate(z) is the complex conjugate of z.
|z+1|^2 = (z+1) * conjugate(z+1)
|z+i|^2 = (z+i) * conjugate(z+i)
Expanding both sides of the equation, we get:
(z+1) * conjugate(z+1) = (z+i) * conjugate(z+i)
By expanding the expressions and simplifying the equation, we have:
(z+1) * (z+1)^* = (z+i) * (z+i)^*
Simplifying further, we get:
(z+1) * (z+1)^* = (z+i) * (z-i)
Now, let's multiply and simplify:
(z^2 + 2z + 1) * (z - i) = (z+i) * (z-i)
Expanding both sides, we have:
z^3 - iz^2 + 2z^2 - 2iz + z - i = z^2 - i^2
Simplifying the right side of the equation, we have:
z^2 - z + 1 = 0
This is a quadratic equation in terms of z, which can be solved using the quadratic formula:
z = (-(-1) ± sqrt((-1)^2 - 4(1)(1))) / (2(1))
Simplifying, we get:
z = (1 ± sqrt(-3))/2
From the solutions above, we can see that the locus of points for |z+1| = |z+i| is the set of points that lie equidistant from -1 and -i. In other words, it is the perpendicular bisector of the line segment connecting -1 and -i in the complex plane.
(b) Re z = |z-1|
To find the locus of points that satisfy this equation, let's first simplify the expression.
Re z = |z-1|
Since Re z represents the real part of z, and |z-1| represents the magnitude (or absolute value) of z-1, the given equation is essentially saying that the real part of z is equal to the distance between z and 1.
Now, let's analyze the equation geometrically. The locus of points that satisfy Re z = |z-1| is the set of all points such that the distance between each point and the point 1 is equal to the real part of that point.
Graphically, this locus represents the subset of the complex plane that lies along a vertical line segment connecting the real axis and the point 1.