Let the product of ω₁ = (8 + i) θ + (7 + 4 i) θ$\omega_1 = (8 + \mathrm{i}) \sin \theta + (7 + 4 \mathrm{i}) \cos \theta$ and ω₂ = (1 + 8 i) θ + (4 + 7 i) θ$\omega_2 = (1 + 8 \mathrm{i}) \sin \theta + (4 + 7 \mathrm{i}) \cos \theta$ be α +iβ$\alpha +\mathrm{i}\beta$, i = √(-1)$\mathrm{i} = \sqrt{-1}$. Let p and q be the maximum and the minimum values of α +β$\alpha +\beta$ respectively.

Core Logic

This implies (x)/(2) = x x = 0$\frac{x}{2} = x \implies x = 0$. There is exactly 1 solution, so α = 1$\alpha = 1$.

Step 1: Simplify complex number z

Thus, z = -π ( (1-√(π)i)(√(π)-i)π + 1 + (√(π)-i)(1-√(π)i)1 + π )$z = -\pi \left( \frac{(1-\sqrt{\pi}i)(\sqrt{\pi}-i)}{\pi + 1} + \frac{(\sqrt{\pi}-i)(1-\sqrt{\pi}i)}{1 + \pi} \right)$

Step 2: Simplify Bracket

Let's expand the terms directly: z = (π)/(4)(-4) [ √(π) - π i - i - √(π)π + 1 + √(π) - i - π i - √(π)1 + π ]$z = \frac{\pi}{4}(-4) \left[ \frac{\sqrt{\pi} - \pi i - i - \sqrt{\pi}}{\pi + 1} + \frac{\sqrt{\pi} - i - \pi i - \sqrt{\pi}}{1 + \pi} \right]$

Step 3: Find beta

For z = 2π i$z = 2\pi i$: |z| = 2π$|z| = 2\pi$ and (z) = (π)/(2)$\arg(z) = \frac{\pi}{2}$.

Step 4: Distance from Line

Distance of point (α, β) = (1, 4)$(\alpha, \beta) = (1, 4)$ from the line 4x - 3y - 7 = 0$4x - 3y - 7 = 0$:

Chapter Mix

Class 11 Maths: Complex Numbers and Quadratic Equations Class 11 Maths: Straight Lines