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

Answer:

The product of z1 and its conjugate is 25.

Step-by-step 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.