Mathematical problems in the complex plane if the complex number z satisfies

1. Choose a. for this questionThe given equation represents the set of points at equal distances to the points (-1,0), (0,1) So the graph is a perpendicular bisector of the line that connects the two points.

2. The given equation represents the set of points whose distance to two fixed points is equal to a fixed length, so it is either an ellipse or a straight segment; And since the distance between the two points is 2, which is equal to the fixed length of 2, the graph is a line break, and the distance from it to the point (2,1) is z=1, and the minimum distance is the root number 2

3. There are two cases here: the positive extension of the ac vector ab is not difficult to obtain c(5,7) from ac=3ab; If the AC vector AB is extended in the opposite direction, then c(-1,-5) is obtained by coordinate operation, so the answer is 5+7i or -1-5i

Fourth, the root number 2; With z'Denotes the conjugate plural of z.

z1+z2|^2=|z1|^2+|z2|^2+z1'z2+z1z2'=2

Thus there is z1'z2+z1z2'=0

z1-z2|^2=|z1|^2+|z2|^2-z1'z2+z1z2'z1z2=1+1-0=2

1.Write the original form as |z+1|=|z-i|, which is geometrically the set of all points with equal distances to point 1 and point i.

So it's a perpendicular bisector of the line between 1 and i, and the answer is A

2.Similar to the previous question, |z+1|+|z-1|The geometric meaning of =2 is the set of all points where the sum of the distances to point -1 and point 1 is 2. With the help of the graph, it is not difficult to see that this is the line segment [-1,1].

So the minimum value of point-to-point (2+i) on the segment is reached at point 1 at a distance of |-1-i|=√2

3.Consider 2 possible C-points in 2 directions on the AB extension. The complex number corresponding to point c is 5+7i or -1-5i

4.|z1|=|z2|=1 means that z1 and z2 are both on the unit garden|z1+z2|= 2, which means that the distance from z1 to -z2 is 2, that is, z1o-z2 is 90 degrees, then the angle of z1oz2 is also 90 degrees, so |z1-z2|=√2

1. |z+1|denotes the distance from point z to -1+0i, |z-i|It represents the distance from z to 0+i, so this means that the distance difference to the two points is zero, so it is the middle perpendicular line of the two-point line segment, choose a

2.According to the meaning of the first question, z represents the line segment connected by (-1,0) and (1,0), that is, the point of the line segment is the closest point to 2+i, so it is 2

3.Because C is on the extension line of AB, it can only be 5+7I4You've studied physics, and there's a graph about forces, |z1+z2|For example, it is the resultant force of forces z1 and z2, and |z1-z2|It's like the difference between two forces.

Click here to draw the figure, and the direct result is: 2

No, don't put the exam questions here, you can ask your teacher.

z+3|+|z-3|=10, this trajectory represents the point z(x,y) to (-3,0), and the sum of the distances from (3,0) is 10, which means that the focal coordinates are f(-3,0),f'(3,0).

In-plane with two fixed points f, f'The sum of distances is equal to the constant 2a(2a>|ff'|The trajectory of the moving point p is called an ellipse. i.e.: PF + PF'│=2a）

By |zf|+|zf'|= 2a = 10, solution a = 5 taste Bai jealousy fat de mixed spring full of Shanghai.

Focal length|ff'|=2c=6,c=3

b = a -c = 5 -3 = 16, so the trajectory equation for the point z is x 25 + y 16 = 1

z-1+i|denotes the distance from z to point a(1-i), |z-i-3|denotes the distance from z to b(i+3), and since the two are equal, (1) denotes the perpendicular bisector of the line segment ab.

2) Let z=x+yi, z*z =x 2+y 2, z+z =2x, so x 2+y 2=2x, simplified to (x-1) 2+y 2=1, is a circle, the center of the circle is (1, 0) and the radius is 1

In the complex plane, |z1-z2|Indicates the distance between points z1 and z2. Thereby.

a(0,1),b(0,-1).The equation represents the perpendicular bisector of the line segment AB.

f1(-2,0),f2(2,0).The equation represents the line segment f1f2

f1(0,-5),f2(0,5).The equation represents the lower branch of the hyperbola with f1 and f2 as the focus and the real axis length of 8.

You can try it yourself.

1, if the plural z satisfies |z|1, then arg(z+2i). Value range.

Solution: The image of z=1 is a circle with the center of the circle as the origin and the radius of 1.

So: the z+2i image is a circle with a center of (0,2) and a radius of 1.

So: arg(z+2i) minimum = (2)-(6) = 3arg(z+2i) maximum = (2)+(6)=2 32, known equation |z-i|+|z+2i|=a.

Eccentricity. is an ellipse of 3 4, then the code dust value of the real number a is solution: |z-i|+|z+2i|=a represents the distance from (x,y) to (0,1) and to (0,-2) and =a, which is a fixed value.

So: c=|1-(-2)|/2=3/2

Let's take the elliptic standard equation in the branch"a"Write a

Then: a=a 2

Eccentricity = c a = (3 2) (a 2) = 3 a = 3 4 so: a = 4

You can let z=x+iy, then (x+1) 2+(y+1) 2=1, and then use the following words to say that one of the columns is called x,y, and z is obtained.

Answer: z=2 (1-i).

2(1+i)/[(1-i)(1+i)]

2(1+i)/2

1+i z-pull (conjugation of z) = 1-i

The corresponding point is (1,-1), which is located in the fourth quadrant.

z-(1-i)|=|z-(i+3)|, so the complex number z represents the set of points at equal distances to point 1-i (coordinates (1,-1)) and points 3+i (coordinates (3,1)) on the complex plane, i.e., the perpendicular bisector of points (1,-1) and points ().

The same |z-i|=|z-3|It represents the set of points at equal distances to (0,1) and (3,0), and is the perpendicular bisector of these two points.

z-i|=|z-3|It means that the distance from point z to point (0,1) on the plane is equal to the distance from point (3,0), and the distance to two points is equal to prove that point z is on the vertical bisector of the line segment formed by these two points.

The second condition is a meaning.

Let a be a plural, then |z-a|=k represents the set of points on the complex plane at a distance k from point a, which is a circle to be exact.

z-i| = |z-3|Represents the set of points on the complex plane that are at equal distances to point i (represented by coordinates (0,1) on the complex plane) and point 3 (represented by coordinates (3,0) on the complex plane) (think, what does this mean?). Isn't the set of points with the same distance to two points the middle perpendicular line connecting these two points? The meaning of the next representation is similar.

z=a+bi

4a+4bi+2a+2bi=6a+6bi

a = root number 3 2 b = 1 6

Modulus = root number type, this root number 3 Bu Hongxun 2 2 + 1 6 2 = root number 7 3