answer:To determine the necessary relationship among ( a, b, c ) such that the following system has a real solution: [ a x + b y = c z, ] [ a sqrt{1 - x^{2}} + b sqrt{1 - y^{2}} = c sqrt{1 - z^{2}}, ] # Step-by-step Process: 1. **Reformulating the Equations:** Let's denote ( d = -c ). By substituting ( d ) into the system, the equations become: [ begin{gathered} a x + b y + d z = 0, a sqrt{1 - x^2} + b sqrt{1 - y^2} + d sqrt{1 - z^2} = 0. end{gathered} ] 2. **Constraints on the Variables:** For the equations to have a real solution, ( x, y, z ) must lie within ([-1, 1]): [ -1 leq x, y, z leq 1. ] 3. **Trigonometric Substitution:** Such values ( x, y, z ) can be expressed using angles (alpha, beta, gamma) such that: [ cos alpha = x, quad cos beta = y, quad cos gamma = z, ] where ( 0 leq alpha, beta, gamma leq pi ). 4. **Vector Representation:** Introduce the following vectors: [ begin{aligned} mathbf{a} &= (a cos alpha, a sin alpha), mathbf{b} &= (b cos beta, b sin beta), mathbf{d} &= (d cos gamma, d sin gamma). end{aligned} ] 5. **Vector Addition Interpretation:** Using these vectors, the system of equations translates to: [ mathbf{a} + mathbf{b} + mathbf{d} = mathbf{0}. ] For this vector equation to hold, the vectors (mathbf{a}, mathbf{b}, mathbf{d}) must form a closed triangle. Thus, their magnitudes must satisfy the triangle inequalities: [ begin{aligned} |a| &leq |b| + |c|, |b| &leq |a| + |c|, |c| &leq |a| + |b|. end{aligned} ] 6. **Sign Condition:** Additionally, from the requirement ( a sqrt{1 - x^2} + b sqrt{1 - y^2} + d sqrt{1 - z^2} = 0 ), and the fact that square roots yield non-negative results, not all coefficients (a, b, d) can have the same sign. Consequently, at least one of (a) or (b) must share the same sign as ( -d = c ). # Conclusion: The necessary conditions for (a, b, c) are: 1. They must satisfy the triangle inequalities: [ begin{aligned} |a| &leq |b| + |c|, |b| &leq |a| + |c|, |c| &leq |a| + |b|. end{aligned} ] 2. At least one of (a) or (b) must have the same sign as (c). [ boxed{text{These conditions ensure the system has a real solution.}} ]

answer:Let's denote the number of sprockets per hour that Machine A produces as A, for Machine B as B, and for Machine C as C. From the information given, we can set up the following relationships: 1. Machine B produces 10% more sprockets per hour than Machine A: B = A + 0.10A B = 1.10A 2. Machine B produces 20% more sprockets per hour than Machine C: B = C + 0.20C B = 1.20C 3. Machine A takes 10 hours longer than Machine B to produce 990 sprockets: 990/A = 990/B + 10 4. Machine C takes 5 hours less than Machine A to produce 990 sprockets: 990/C = 990/A - 5 Now, let's solve these equations step by step. From equation 1: B = 1.10A From equation 2: B = 1.20C Since both expressions equal B, we can set them equal to each other: 1.10A = 1.20C Now, let's express C in terms of A using the relationship between A and C from equation 4: 990/C = 990/A - 5 C = 990 / (990/A - 5) C = A / (1 - 5A/990) Substitute C in the equation 1.10A = 1.20C: 1.10A = 1.20(A / (1 - 5A/990)) Now, let's solve for A: 1.10A = 1.20A / (1 - 5A/990) 1.10A(1 - 5A/990) = 1.20A 1.10 - 5.5A/990 = 1.20 1.10 - 1.20 = 5.5A/990 -0.10 = 5.5A/990 -0.10 * 990 = 5.5A -99 = 5.5A A = -99 / 5.5 A = -18 Since the number of sprockets per hour cannot be negative, there must be a mistake in our calculations. Let's re-evaluate the steps. We made a mistake in the simplification step. Let's correct it: 1.10A(1 - 5A/990) = 1.20A 1.10 - 5.5A/990 = 1.20 -0.10 = -5.5A/990 0.10 = 5.5A/990 0.10 * 990 = 5.5A 99 = 5.5A A = 99 / 5.5 A = 18 So, Machine A produces boxed{18} sprockets per hour.

answer:Given that the parabola passes through points A(2,0) and B(-1,0), and intersects the y-axis at point C with OC=2, we can determine the coordinates of point C as either (0,2) or (0,-2) due to the distance from the origin. The general form of a parabola that passes through points A and B can be expressed as y=a(x-2)(x+1). This is because the roots of the parabola are at x=2 and x=-1, which correspond to the x-coordinates of points A and B. First, let's consider the case where C=(0,2): - Substituting (0,2) into the equation gives us 2=a(0-2)(0+1) Rightarrow 2=a(-2) Rightarrow a=-1. - Therefore, the equation of the parabola becomes y=-1(x-2)(x+1), which simplifies to y=-x^2+x+2. Next, considering the case where C=(0,-2): - Substituting (0,-2) into the equation gives us -2=a(0-2)(0+1) Rightarrow -2=a(-2) Rightarrow a=1. - Therefore, the equation of the parabola becomes y=1(x-2)(x+1), which simplifies to y=x^2-x-2. Hence, the analytical expressions of the parabola are y=-x^2+x+2 or y=x^2-x-2. Therefore, the final answer is boxed{y=-x^2+x+2 text{ or } y=x^2-x-2}.

answer:Let's denote the number of goldfish in the first fish tank as G. According to the information given, the third fish tank has a third of the number of fish in the second fish tank, and there are 10 fish in the third fish tank. This means that the second fish tank has 3 times the number of fish in the third fish tank: Number of fish in the second tank = 3 * Number of fish in the third tank Number of fish in the second tank = 3 * 10 Number of fish in the second tank = 30 The second fish tank contains twice as many fish as the first tank. Therefore, the first fish tank has half the number of fish as the second tank: Number of fish in the first tank = 1/2 * Number of fish in the second tank Number of fish in the first tank = 1/2 * 30 Number of fish in the first tank = 15 The first fish tank contains some goldfish (G) and 8 beta fish. So the total number of fish in the first tank is the sum of the goldfish and the beta fish: Number of fish in the first tank = G + 8 We already know that the first fish tank has 15 fish in total, so we can set up the equation: G + 8 = 15 Now, we can solve for G (the number of goldfish): G = 15 - 8 G = 7 There are boxed{7} goldfish in the first fish tank.