Question

REALLY NEED HELP!!!
see picture*****

Answer 1

11)
`r=4`

12)
`z_3+z_1=A`

13a)
`\bar z_1=3-2i` and
`\bar z_2=-5+3i`

13b)
`\bar {z_1z_2}=-9-19i`

Explanation:

11. The given equation in rectangular coordinates is
`x^2+y^2=16`

To convert to polar form, use the relation
`x^2+y^2=r^2`

This implies that:

`\implies r^2=16`

`r=√(16)`

`r=4`

In polar coordinates,
`r=4` is a circle with radius 4 units.

12. From the diagram;

`z_3=-3-3i`

`z_2=1+i`

`z_1=4+i`

`A=1-2i`

We can see that two arrows are moving in anticlockwise direction to meet A which is moving in the clockwise direction;

`\implies (-3-3i)+(4+i)=1-2i`

`\therefore z_3+z_1=A`

13. The conjugate of the complex number:
`z=a+bi` is
`\bar z=a-bi`

Part a) The given complex numbers are:

`z_1=3+2i` and
`z_2=-5-3i`

The conjugates of these complex numbers are:

`\bar z_1=3-2i` and
`\bar z_2=-5+3i`

Part b) The product of
`z_1=3+2i` and
`z_2=-5-3i` is

`z_1z_2=(3+2i)(-5-3i)`

`z_1z_2=-15-9i-10i-6i^2`

`z_1z_2=-9-19i`

The conjugate of the product is:

`\bar {z_1z_2}=-9-19i`