Final answer:
To find the range of values for the real constant r, we need to solve the two given equations: |z-3-5i| = 2r and arg(z-2) = 3π/4. By solving these equations and simplifying, we find that the range of values for r is (-∞, +∞).
Step-by-step explanation:
To find the range of values for the real constant r, we need to solve the two given equations:
|z-3-5i| = 2r
arg(z-2) = 3π/4
Let's solve each equation separately:
- From the first equation, |z-3-5i| = 2r, we can rewrite it as:
- √((x-3)² + (y-5)²) = 2r
- (x-3)² + (y-5)² = 4r²
From the second equation, arg(z-2) = 3π/4, we can rewrite it as:
- arg((x-2) + yi) = 3π/4
- (y-2)/(x-2) = tan(3π/4)
- Since tan(3π/4) = -1, we have (y-2)/(x-2) = -1
- y-2 = -x+2
- y = -x+4
Substitute y = -x+4 into the first equation:
- (x-3)² + (-x+4-5)² = 4r²
- x² - 6x + 9 + (x-1)² = 4r²
- 2x² - 8x + 10 = 4r²
- x² - 4x + 5 = 2r²
Since r is a real constant, 2r² is also a real number. Therefore, x² - 4x + 5 must be non-negative:
- x² - 4x + 5 ≥ 0
- (x-2)² + 1 ≥ 0
- This inequality holds for all real values of x.
Hence, the exact range of values for the real constant r is (-∞, +∞).