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

Question

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

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

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

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

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

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

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

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

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:

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

Rewrite the equation separating real and imaginary parts:

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

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

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:

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

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

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

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

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