answer:To solve this problem, let's follow the given solution closely but break it down into more detailed steps: 1. **Express z and overline{z} in terms of x and y:** Given z = x + yi, where x, y in mathbb{R}, the conjugate of z, denoted as overline{z}, is x - yi. 2. **Translate the given condition into an equation:** The condition |overline{z} + 4 - 2i| = 1 can be rewritten using overline{z} = x - yi, resulting in |x - yi + 4 - 2i| = 1. Simplifying inside the modulus gives |(x + 4) - (y + 2)i| = 1. 3. **Apply the modulus formula for complex numbers:** The modulus of a complex number a + bi is sqrt{a^2 + b^2}. Applying this to our equation, we get sqrt{(x + 4)^2 + (y + 2)^2} = 1. Squaring both sides to remove the square root yields (x + 4)^2 + (y + 2)^2 = 1^2 = 1. 4. **Interpret the equation geometrically:** The equation (x + 4)^2 + (y + 2)^2 = 1 represents a circle in the complex plane with center O(-4, -2) and radius r = 1. 5. **Find the distance from z to i:** The distance from any point z = x + yi to the point i (which corresponds to (0,1) in the complex plane) is given by |z - i| = sqrt{x^2 + (y - 1)^2}. 6. **Determine the minimum distance from the circle to point P(0,1):** The minimum distance from any point on the circle to P(0,1) is the distance from the center of the circle O(-4, -2) to P(0,1) minus the radius of the circle. This can be calculated as follows: - Distance |OP| = sqrt{(-4 - 0)^2 + (-2 - 1)^2} = sqrt{16 + 9} = sqrt{25} = 5. - Therefore, the minimum value of |z - i| is |OP| - r = 5 - 1 = 4. Thus, the minimum value of |z - i| is boxed{4}, which corresponds to option boxed{text{A}}.

answer:(1) Since f(x)=ax^{3}+bx+c, we have f'(x)=3ax^{2}+b. Given that f(x) reaches an extreme value c-4 at x=1, we have the system of equations begin{cases} f(1)=c-4 f'(1)=0 end{cases}, which leads to begin{cases} a+b+c=c-4 3a+b=0 end{cases}. Solving the system yields begin{cases} a=2 b=-6 end{cases}. (2) Since y=f(x) is an odd function on R, we have f(-x)=-f(x), which implies a(-x)^{3}+b(-x)+c=-(ax^{3}+bx+c). From this equation, we deduce that c=0. Hence, f(x)=2x^{3}-6x. Now, let's find the derivative: f'(x)=6x^{2}-6=6(x+1)(x-1). Setting f'(x)=0, we obtain the critical points x=-1 and x=1. For the interval (-2,0), we consider x=-1. - When x in (-2,-1), f'(x) > 0, - When x in (-1,0), f'(x) < 0. Therefore, the function f(x) has a maximum value at x=-1. The maximum value is f(-1) = boxed{4}. There is no minimum value in this interval.

answer:Using the identity (x^2 - y^2 = (x + y)(x - y)): 1. Plug the given values into the formula: [ x^2 - y^2 = frac{9}{17} times frac{1}{51} ] 2. Calculate the product: [ x^2 - y^2 = frac{9 times 1}{17 times 51} = frac{9}{867} ] 3. Simplify the fraction (if possible): Note that 867 is actually (3 times 17 times 17), and doesn't simplify with 9. Conclusion with boxed answer: [ boxed{frac{9}{867}} ]

answer:To find the length of the rectangular garden, we can use the formula for the perimeter of a rectangle: Perimeter (P) = 2 * (Length (L) + Breadth (B)) Given that the perimeter (P) is 900 m and the breadth (B) is 190 m, we can plug these values into the formula and solve for the length (L): 900 m = 2 * (L + 190 m) First, divide both sides of the equation by 2 to isolate the term with the length: 900 m / 2 = L + 190 m 450 m = L + 190 m Now, subtract 190 m from both sides to solve for the length (L): 450 m - 190 m = L 260 m = L Therefore, the length of the rectangular garden is boxed{260} meters.