the question says to sketch the set of points in the complex plane determined by the follow.
absvalue(z-(1-i))=1
this is my idea:
absvalue(z-(1-i))=
absvalue(z)-absvalue(1-i)
absvalue(z)-sqrt(2)=1
absvalue(z)=1-sqrt(2)
z=(1-sqrt(2))^2=1-2*sqrt(2)+2=3-2*sqrt(2)=
x^2+y^2
then what?
To sketch the set of points in the complex plane determined by the equation |z-(1-i)| = 1, we can follow a step-by-step process:
1. Start by rewriting the equation in its absolute value form:
|z-(1-i)| = 1
2. The absolute value of a complex number z = x + yi is defined as:
|z| = sqrt(x^2 + y^2)
Applying this definition to our equation, we get:
sqrt((x - 1)^2 + (y - (-1))^2) = 1
3. Simplify the equation:
(x - 1)^2 + (y + 1)^2 = 1
Expanded form:
x^2 - 2x + 1 + y^2 + 2y + 1 = 1
Rearrange the equation:
x^2 + y^2 - 2x + 2y + 1 = 0
4. Now, we have the equation in the standard form of a circle. Comparing it with the general equation of a circle:
(x - h)^2 + (y - k)^2 = r^2
We can see that the center of the circle is (h, k) = (1, -1), and the radius is r = 1.
5. Using this information, we can sketch the circle with center (1, -1) and radius 1 on the complex plane.
- The center (1, -1) represents the point 1 - i in the complex plane (corresponding to x and y coordinates).
- The circle's radius is 1, which means all the points on the circle have an absolute value of 1 from the center.
- Plotting the circle on the complex plane, you will have a circle centered at (1, -1) with a radius of 1.
Note: The equation you obtained in your idea, z = 3 - 2*sqrt(2), does not represent the set of points required to be sketched. It only represents a single point in the complex plane. The correct method involves sketching a circle.