If z=(1)/(2)-2i is such that |z+1|= z+ (1+i),i=-1 and , R, then + is equal to

Question

If $z=\frac{1}{2}-2i$ is such that $|z+1|=\alpha z+\beta (1+i),i=\sqrt{-1}$ and $\alpha ,\beta \in R$, then $\alpha +\beta $ is equal to

Accepted Answer

To begin with, let's analyze the given equation:

$|z+1|=\alpha z+\beta (1+i)$

First, we compute the modulus of the left side:

Let's take the given value of z,

$z=\frac{1}{2}-2i$

Now, we find $z+1$:

$z+1=(\frac{1}{2}-2i)+1=\frac{3}{2}-2i$

Then, the modulus of $z+1$ is calculated as follows:

$|z+1|=|\frac{3}{2}-2i|=\sqrt{{(\frac{3}{2})}^{2}+(-2{)}^{2}}$

$|z+1|=\sqrt{\frac{9}{4}+4}$

$|z+1|=\sqrt{\frac{9}{4}+\frac{16}{4}}$

$|z+1|=\sqrt{\frac{25}{4}}$

$|z+1|=\frac{5}{2}$

Now we need to equate the modulus to the right-hand side of the equation and solve for $\alpha $ and $\beta $. Let's rewrite the equation:

$\frac{5}{2}=\alpha z+\beta (1+i)$

Substitute $z$ with its value:

$\frac{5}{2}=\alpha (\frac{1}{2}-2i)+\beta (1+i)$

Rewrite the equation separating real and imaginary parts:

$\frac{5}{2}=\alpha (\frac{1}{2})-2\alpha i+\beta +\beta i$

$\frac{5}{2}=(\alpha \frac{1}{2}+\beta )+(-2\alpha +\beta )i$

For the above equality to hold, both real and imaginary parts must be equal.

Equating real parts:

$\alpha \frac{1}{2}+\beta =\frac{5}{2}$

Equating imaginary parts:

$-2\alpha +\beta =0$

We now have a system of two linear equations:

1) $\frac{\alpha }{2}+\beta =\frac{5}{2}$

2) $-2\alpha +\beta =0$

Let's solve the system by isolating $\beta $ from the second equation and then substituting it into the first one:

$\beta =2\alpha $

Now substitute $\beta $ in the first equation:

$\frac{\alpha }{2}+2\alpha =\frac{5}{2}$

$\alpha (\frac{1}{2}+2)=\frac{5}{2}$

$\alpha (\frac{1}{2}+\frac{4}{2})=\frac{5}{2}$

$\alpha (\frac{5}{2})=\frac{5}{2}$

To find the value of $\alpha $, we divide both sides by $\frac{5}{2}$:

$\alpha =1$

Now, we use the value of $\alpha $ to find $\beta $:

$\beta =2\alpha $

$\beta =2\cdot 1$

$\beta =2$

Finally, we add both $\alpha $ and $\beta $ to find $\alpha +\beta $:

$\alpha +\beta =1+2=3$

The value of $\alpha +\beta $ is 3.

So, the correct answer is Option C) 3.

Solutions · 1