If denotes the number of solutions of |1-i|^x=2^x (x R) and is [a value defined by a furth…

Question

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ ($x\in\mathbb R$) and $\beta$ is [a value defined by a further condition not preserved in the source], then the distance of the point $(\alpha,\beta)$ from the line $4x-3y=7$ is \_\_\_\_

Answer: 3

Accepted Answer

Since $|1-i|=\sqrt2$, the equation $|1-i|^x=2^x$ becomes $2^{x/2}=2^x \implies \dfrac x2=x \implies x=0$. This is the only solution, so $\alpha=1$.

*(The definition of $\beta$ was not preserved in the extracted source text. Using the stated answer, the distance formula $\dfrac{|4\alpha-3\beta-7|}{5}=3$ with $\alpha=1$ gives $|{-3\beta-3}|=15$, consistent with $\beta=4$, though this value of $\beta$ could not be independently confirmed from the garbled source.)*

Solutions · 1