To solve the problem, we need to understand and analyze the given curve and the circle it affects. The curve is defined by the equation \(z(1 + i) + \overline{z(1 - i)} = 4\). Let's start by breaking down the steps:
Let \(z = x + iy\) where \(x, y \in \mathbb{R}\). The complex conjugate \(\overline{z}\) is \(x - iy\). Substituting into the curve equation:
\((x + iy)(1 + i) + (x - iy)(1 - i) = 4\)
Simplify both products:
\((x + iy)(1 + i) = x(1) + xi + iy(1) + i^2y = x + xi + iy - y = (x - y) + i(x + y)\)
\((x - iy)(1 - i) = x(1) - xi - iy(1) + i^2y = x - xi - iy + y = (x + y) - i(x - y)\)
Sum the real and imaginary parts:
\((x - y) + i(x + y) + (x + y) - i(x - y) = 4\)
Combine like terms:
\((x - y) + (x + y) + i[(x + y) - (x - y)] = 4\)
This results in \(2x + 2iy = 4\).
Real part, \(2x = 4\) gives \(x = 2\).
Imaginary part, \(2iy = 0\) gives \(y = 0\).
The line through \(z = 2\) is vertical in terms of our circle \(|z - 3| \leq 1\), which is a circle centered at \(z = 3\) with radius 1.
Geometrically, the line \(x = 2\) divides the circle into two regions, each of area \(\frac{\pi}{2}\). The small segment's area needs the length of the chord created by \(x = 2\).
The circle's center-radius form \((x - 3)^2 + y^2 = 1\). Substituting \(x = 2\), solving for \(y\):
\((2 - 3)^2 + y^2 = 1\) ⟹ \(y^2 = 0\) ⟹ \(y = 0\).
The chord is horizontal at the center at maximum y-value via perpendicular height from center to line \((FS = 1)\), creating a right triangle. The area of the segment \(\bar{G} = \frac{\pi}{4}\).
Thus, \(\alpha = \bar{G} + 1\), \(\beta = \frac{\pi}{2}-\bar{G}\).
Area difference \(|\alpha - \beta|\) yields:
\(|(\frac{\pi}{4} + 1) - \frac{\pi}{2}| = |1 + \frac{\pi}{4} - \frac{\pi}{2}| = |1 - \frac{\pi}{4}|\), which chose as:
\( {1 + \frac{\pi}{4}} \).
If $ \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p $, then $ 96 \log_e p $ is equal to _______
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to: