Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, its modulus is $|z| = \sqrt{x^2 + y^2}$.
• Argument of a Complex Number: The argument, $\arg(z)$, is the angle $z$ makes with the positive real axis. For $z=iy$ with $y>0$, $\arg(z) = \frac{\pi}{2}$.
• Distance of a Point from a Line: The distance $D$ of a point $(x_1, y_1)$ from a line $Ax + By + C = 0$ is $D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.

Step-by-Step Solution

Part 1: Finding the value of $\alpha$

Step 1: Calculate the modulus of $1-i$

We need to find $|1-i|$ to simplify the given equation. $|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$ Explanation: We use the definition of the modulus of a complex number.

Step 2: Substitute and simplify the equation

Substitute $|1-i| = \sqrt{2}$ into the equation $|1-i|^x = 2^x$. $(\sqrt{2})^x = 2^x$ Rewrite $\sqrt{2}$ as $2^{1/2}$: $(2^{1/2})^x = 2^x$ $2^{x/2} = 2^x$ Explanation: Expressing both sides with the same base allows us to compare exponents.

Step 3: Solve for $x$

Since the bases are equal, the exponents must be equal: $\frac{x}{2} = x$ $x = 2x$ $x - 2x = 0$ $-x = 0$ $x = 0$ Therefore, $\alpha = 1$. Explanation: Solving the equation for $x$ gives the only solution.

Part 2: Finding the value of $\beta$

Step 1: Simplify $(1+i)^4$

We simplify this term first to reduce the complexity of the expression for $z$. $(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i$ $(1+i)^4 = ((1+i)^2)^2 = (2i)^2 = 4i^2 = -4$ Explanation: We use the binomial expansion and the property $i^2 = -1$.

Step 2: Simplify $\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}$

We simplify each fraction separately and then add them. $\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i} = \frac{(1-\sqrt{\pi} i)(\sqrt{\pi}-i)}{(\sqrt{\pi}+i)(\sqrt{\pi}-i)} = \frac{\sqrt{\pi} - i - \pi i - \sqrt{\pi}}{\pi + 1} = \frac{-i - \pi i}{\pi + 1} = \frac{-i(1+\pi)}{1+\pi} = -i$ $\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i} = \frac{(\sqrt{\pi}-i)(1-\sqrt{\pi}i)}{(1+\sqrt{\pi} i)(1-\sqrt{\pi}i)} = \frac{\sqrt{\pi} - \pi i - i - \sqrt{\pi}}{1+\pi} = \frac{-i - \pi i}{1+\pi} = \frac{-i(1+\pi)}{1+\pi} = -i$ Therefore, $\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i} = -i + (-i) = -2i$ Explanation: We multiply the numerator and denominator of each fraction by the conjugate of the denominator to rationalize it.

Step 3: Simplify the expression for $z$

$z = \frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right]$ $z = \frac{\pi}{4}(-4)(-2i) = 2\pi i$ Explanation: We substitute the simplified values from the previous steps.

Step 4: Calculate $|z|$ and $\arg(z)$

$|z| = |2\pi i| = \sqrt{0^2 + (2\pi)^2} = 2\pi$ $\arg(z) = \frac{\pi}{2}$ Explanation: We use the definitions of modulus and argument for a purely imaginary number.

Step 5: Calculate $\beta$

$\beta = \frac{|z|}{\arg(z)} = \frac{2\pi}{\frac{\pi}{2}} = \frac{2\pi \cdot 2}{\pi} = 4$ Explanation: We substitute the values of $|z|$ and $\arg(z)$ into the definition of $\beta$.

Part 3: Calculate the Distance

Step 1: State the point and the line equation

We have the point $(\alpha, \beta) = (1, 4)$ and the line $4x - 3y = 7$, which can be written as $4x - 3y - 7 = 0$.

Step 2: Apply the distance formula

$D = \frac{|4(1) - 3(4) - 7|}{\sqrt{4^2 + (-3)^2}} = \frac{|4 - 12 - 7|}{\sqrt{16 + 9}} = \frac{|-15|}{\sqrt{25}} = \frac{15}{5} = 3$ Explanation: We substitute the values into the distance formula and calculate the result.

Common Mistakes & Tips

• Careless arithmetic, especially with signs, can lead to errors. Double-check each calculation.
• When simplifying complex fractions, multiplying by the conjugate is a reliable method.
• Remember the correct formula for the distance between a point and a line.

Summary

By carefully simplifying the given expressions and applying the relevant formulas, we found that $\alpha = 1$, $\beta = 4$, and the distance of the point $(\alpha, \beta)$ from the line $4x - 3y = 7$ is $3$. There seems to be an error in the question, as the correct distance is 3, not 2 as stated in the provided answer.

Final Answer

The final answer is $\boxed{3}$.