Question

What is the product of z1 and its conjugate?

Answer 1

From the plot, we see that
`z_1=-4-3i`. Its conjugate would be
`\bar{z_1}=-4+3i`, so that the product of the two is

`z_1\bar{z_1}=(-4-3i)(-4+3i)=16-9i^2=16+9=25`

More generally, note that if
`z=x+yi`, then

`z\bar z=(x+yi)(x-yi)=x^2+y^2=|z|^2`

Answer 2

The product of z1 and its conjugate is 25.

Explanation:

In the given graph x-axis represents the real axis and y-axis represents the imaginary axis.

The end point of z1 are (0,0) and (-4,-3). So, the complex number z1 is defined as

`z_1=x+iy=-4-3i`

The conjugate of z1 is

`\overline {z_1}=x-iy=-4+3i`

The product of z1 and its conjugate is

`z_1\overline {z_1}=(-4-3i)(-4+3i)`

`z_1\overline {z_1}=-4(-4+3i)-3i(-4+3i)`

`z_1\overline {z_1}=16-12i+12i-(3i)^2`

`z_1\overline {z_1}=16-9(i)^2`

`z_1\overline {z_1}=16-9(-1)`
`[\because i^2=-1]`

`z_1\overline {z_1}=16+9=25`

Therefore the product of z1 and its conjugate is 25.