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Given that i is an imaginary unit, determine the quadrant in the complex plane corresponding to the point represented by frac{3-i}{2+i}.A: First quadrantB: Second quadrantC: Third quadrantD: Fourth quadrant

💡To determine the quadrant in the complex plane corresponding to the point represented by (frac{3 - i}{2 + i}), we can follow these steps:1. **Simplify the Complex Fraction:** Multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary unit from the denominator. [ frac{3 - i}{2 + i} = frac{(3 - i)(2 - i)}{(2 + i)(2 - i)} ]2. **Expand the Numerator and Denominator:** Use the distributive property to expand both the numerator and the denominator. [ text{Numerator: } (3 - i)(2 - i) = 6 - 3i - 2i + i^2 = 6 - 5i - 1 = 5 - 5i ] [ text{Denominator: } (2 + i)(2 - i) = 4 - i^2 = 4 - (-1) = 5 ]3. **Simplify the Expression:** Divide both the real and imaginary parts of the numerator by the denominator. [ frac{5 - 5i}{5} = 1 - i ]4. **Identify the Quadrant:** The simplified expression (1 - i) corresponds to the point ((1, -1)) in the complex plane. Since the real part (1) is positive and the imaginary part (-1) is negative, the point lies in the fourth quadrant.Therefore, the point (frac{3 - i}{2 + i}) is located in the fourth quadrant of the complex plane.

If a positive integer x divided by (9p) is an odd prime number, where p is a prime number, the least value of x is 54. What is the least value of x?

💡First, I need to understand the problem: I'm given that when a positive integer ( x ) is divided by ( 9p ), where ( p ) is a prime number, the result is an odd prime number. Additionally, it's stated that the least value of ( x ) is 54. My goal is to determine the least value of ( x ).To approach this, I'll start by setting up the equation based on the given information:[frac{x}{9p} = text{odd prime number}]Given that ( x ) is 54, I substitute this value into the equation:[frac{54}{9p} = text{odd prime number}]Simplifying the left side by dividing 54 by 9:[frac{6}{p} = text{odd prime number}]Since ( p ) is a prime number and the result must be an odd prime number, I consider the smallest odd prime number, which is 3. I set up the equation:[frac{6}{p} = 3]Solving for ( p ):[p = frac{6}{3} = 2]However, ( p ) must be a prime number, and 2 is the only even prime number. Since the result must be an odd prime number, ( p ) cannot be 2. The next smallest odd prime number is 3. I check if ( p ) can be 3:[frac{6}{3} = 2]This doesn't work because 2 is not an odd prime number. I realize there might be a mistake in my initial assumption. I revisit the problem and consider that the least value of ( x ) might not be 54 but rather needs to be determined.I set up the equation again, letting ( frac{x}{9p} = q ), where ( q ) is an odd prime number. Therefore:[x = 9p times q]To find the least value of ( x ), I choose the smallest prime numbers for ( p ) and ( q ). The smallest prime number is 2, and the smallest odd prime number is 3. Substituting these values:[x = 9 times 2 times 3 = 54]However, this results in ( frac{54}{9 times 2} = 3 ), which is an odd prime number. This seems to satisfy the conditions. But earlier, I thought there was an inconsistency. I realize that my confusion arose from misinterpreting the given information. The least value of ( x ) is indeed 54, as it satisfies the condition when ( p = 2 ) and ( q = 3 ).Therefore, the least value of ( x ) is 54.

Consider three polynomials, P(z), S(z), and T(z) with real coefficients, where P(z) has degree 3 with a constant term of 4, S(z) has degree 4 with a constant term of 5, and T(z) has degree 7 with a constant term of 20. Determine the minimum number of distinct complex solutions to the equation P(z) cdot S(z) = T(z).(A) 0(B) 1(C) 2(D) 3(E) 7

💡Okay, so I have this problem here about three polynomials: P(z), S(z), and T(z). They all have real coefficients, which is important. Let me break down what each polynomial is like.First, P(z) is a cubic polynomial, meaning its degree is 3. It has a constant term of 4. So, if I write it out, it would look something like P(z) = a*z^3 + b*z^2 + c*z + 4, where a, b, c are real coefficients, and a is not zero because it's a cubic.Next, S(z) is a quartic polynomial, so its degree is 4. It has a constant term of 5. So, similarly, S(z) = d*z^4 + e*z^3 + f*z^2 + g*z + 5, with d, e, f, g being real coefficients and d not zero.Then, T(z) is a polynomial of degree 7 with a constant term of 20. So, T(z) = h*z^7 + ... + k*z + 20, where h is not zero.The question is asking for the minimum number of distinct complex solutions to the equation P(z) * S(z) = T(z). The options are from 0 to 7, so I need to figure out the least number of solutions possible.Let me think about the degrees first. When I multiply P(z) and S(z), the degrees add up, so 3 + 4 = 7. That means P(z)*S(z) is a degree 7 polynomial, just like T(z). So, in terms of degree, they match up.Now, looking at the constant terms. The constant term of P(z) is 4, and the constant term of S(z) is 5. When I multiply them, the constant term of P(z)*S(z) should be 4*5 = 20, which is exactly the constant term of T(z). So, that matches too.Hmm, so both the degrees and the constant terms match. That suggests that if P(z)*S(z) equals T(z), then their leading coefficients must also match. Let me check that.Suppose P(z) has leading coefficient a, and S(z) has leading coefficient d. Then, the leading term of P(z)*S(z) would be a*d*z^7. On the other hand, T(z) has leading coefficient h. So, for P(z)*S(z) to equal T(z), we must have a*d = h.So, if I choose a, d, and h such that a*d = h, then the leading terms will match. That seems possible. So, in theory, it's possible that P(z)*S(z) = T(z) exactly, meaning that the equation P(z)*S(z) - T(z) = 0 is identically zero. But that would mean that the equation has infinitely many solutions, right? Because every z would satisfy it.But the question is about the minimum number of distinct complex solutions. So, maybe I don't want P(z)*S(z) to equal T(z) exactly. Maybe I want them to differ by something, so that the equation P(z)*S(z) = T(z) has as few solutions as possible.Wait, but if I make P(z)*S(z) - T(z) a non-zero polynomial, then by the Fundamental Theorem of Algebra, it will have as many roots as its degree, counting multiplicities. So, if P(z)*S(z) - T(z) is a degree 7 polynomial, it will have 7 roots in the complex plane, counting multiplicities. But the question is about distinct roots.So, to minimize the number of distinct complex solutions, I need to maximize the multiplicities of the roots. That is, I want as many roots as possible to coincide. The minimal number of distinct roots would be 1 if all roots are the same, but is that possible?Let me think. If P(z)*S(z) - T(z) is a polynomial of degree 7, can it have all roots the same? That would mean it's a polynomial like (z - a)^7 for some complex number a. Is that possible?Well, let's see. If P(z)*S(z) - T(z) = (z - a)^7, then P(z)*S(z) = T(z) + (z - a)^7. But T(z) is a given polynomial of degree 7 with constant term 20. So, if I set a = 0, then (z - 0)^7 = z^7, and T(z) would have to be P(z)*S(z) - z^7. But T(z) has a constant term of 20, and P(z)*S(z) has a constant term of 20 as well. So, P(z)*S(z) - z^7 would have a constant term of 20 - 0 = 20, which matches T(z)'s constant term.But wait, T(z) is given as a specific polynomial, not just any polynomial. So, unless P(z)*S(z) - z^7 equals T(z), which would require that P(z)*S(z) = T(z) + z^7. But T(z) is already a degree 7 polynomial, so adding z^7 would make it degree 7 as well, but with a different leading coefficient.Wait, no. If T(z) is degree 7, and P(z)*S(z) is also degree 7, then P(z)*S(z) - T(z) is a degree 7 polynomial minus another degree 7 polynomial, which could result in a lower degree polynomial if the leading terms cancel out. But in our case, if P(z)*S(z) - T(z) is supposed to be (z - a)^7, which is degree 7, then the leading terms must not cancel out. So, the leading coefficients must be equal.So, if P(z)*S(z) - T(z) = (z - a)^7, then the leading coefficient of P(z)*S(z) must be equal to the leading coefficient of T(z) plus 1, because (z - a)^7 has leading coefficient 1. Wait, no. Let me clarify.If P(z)*S(z) - T(z) = (z - a)^7, then the leading term of P(z)*S(z) is equal to the leading term of T(z) plus the leading term of (z - a)^7. But (z - a)^7 has leading term z^7, so the leading term of P(z)*S(z) would be equal to the leading term of T(z) plus z^7. But that would mean that the leading coefficient of P(z)*S(z) is h + 1, where h is the leading coefficient of T(z). But earlier, we had that the leading coefficient of P(z)*S(z) is a*d, and it must equal h. So, unless h + 1 = h, which is impossible, this can't happen.Therefore, it's impossible for P(z)*S(z) - T(z) to be equal to (z - a)^7 because the leading coefficients would conflict. So, that approach doesn't work.Maybe I need to think differently. Perhaps instead of trying to make P(z)*S(z) - T(z) have all roots the same, I can make it have as few distinct roots as possible, but not necessarily all the same.What's the minimal number of distinct roots a degree 7 polynomial can have? Well, it can have 1 root with multiplicity 7, but as we saw, that might not be possible here. Alternatively, it can have 2 roots, one with multiplicity 6 and another with multiplicity 1, or other combinations.But let's think about the structure of P(z)*S(z) - T(z). Since P(z) and S(z) are polynomials with real coefficients, their product P(z)*S(z) will also have real coefficients. Similarly, T(z) has real coefficients, so P(z)*S(z) - T(z) will have real coefficients.Therefore, any complex roots must come in conjugate pairs. So, if there's a complex root a + bi, there must also be a root a - bi. So, the number of non-real roots must be even. Therefore, the number of distinct complex roots must be even, unless there are real roots.But the question is about distinct complex solutions, which includes both real and non-real roots. So, complex solutions here mean solutions in the complex plane, not necessarily non-real.Wait, actually, in mathematics, "complex solutions" can sometimes mean non-real solutions, but in the context of polynomials with real coefficients, complex solutions are often considered to include both real and non-real solutions, with non-real ones coming in conjugate pairs.But the problem says "distinct complex solutions," so I think it includes all solutions in the complex plane, both real and non-real. So, the minimal number would be considering both.But let's get back. Since P(z)*S(z) - T(z) is a degree 7 polynomial, it must have 7 roots in the complex plane, counting multiplicities. The minimal number of distinct roots would be 1 if all roots are the same, but as we saw earlier, that might not be possible.Alternatively, maybe it can have 2 distinct roots, one with multiplicity 6 and another with multiplicity 1, or something like that.But is that possible? Let's see. Suppose P(z)*S(z) - T(z) = (z - a)^6*(z - b), where a and b are distinct complex numbers. Then, it would have two distinct roots, a and b, with multiplicities 6 and 1 respectively.But can such a polynomial exist given the constraints on P(z), S(z), and T(z)?Alternatively, maybe it's possible to have a polynomial with only one distinct root, but as we saw earlier, that might not work because of the leading coefficient issue.Wait, let me think again. If P(z)*S(z) - T(z) is a degree 7 polynomial, and if it has only one distinct root, then it must be of the form (z - a)^7. But as we saw earlier, that would require the leading coefficient of P(z)*S(z) to be equal to the leading coefficient of T(z) plus 1, which is impossible because the leading coefficient of P(z)*S(z) is fixed by the leading coefficients of P(z) and S(z), and T(z) is given.Therefore, it's impossible for P(z)*S(z) - T(z) to be a perfect 7th power of a linear term. So, the minimal number of distinct roots must be more than 1.What about 2 distinct roots? Let's see. Suppose P(z)*S(z) - T(z) = (z - a)^k*(z - b)^{7 - k}, where k is between 1 and 6. Then, it would have two distinct roots, a and b.Is this possible? Let's see. If I can choose P(z) and S(z) such that their product minus T(z) factors into such a form, then yes.But how?Alternatively, maybe I can think about specific choices for P(z) and S(z) that make P(z)*S(z) - T(z) have as few distinct roots as possible.For example, suppose I choose P(z) = z^3 + 4 and S(z) = z^4 + 5. Then, P(z)*S(z) = z^7 + 5z^3 + 4z^4 + 20. If I set T(z) = z^7 + 4z^4 + 5z^3 + 20, then P(z)*S(z) - T(z) = 0, which would mean that every z is a solution, but that's not helpful because we want the minimal number of solutions.Wait, but in this case, P(z)*S(z) = T(z), so the equation P(z)*S(z) = T(z) is always true, meaning every complex number is a solution. But the problem is asking for the minimal number of distinct complex solutions, so this case is not helpful because it gives infinitely many solutions.Instead, I need to choose P(z) and S(z) such that P(z)*S(z) - T(z) is a non-zero polynomial of degree 7 with as few distinct roots as possible.Alternatively, maybe I can make P(z)*S(z) - T(z) have a multiple root with high multiplicity.Wait, but as we saw earlier, making it a single root with multiplicity 7 is impossible because of the leading coefficient conflict. So, maybe the next best thing is to have two distinct roots, one with multiplicity 6 and another with multiplicity 1.But is that possible?Alternatively, perhaps I can make P(z)*S(z) - T(z) have a root at z = 0 with multiplicity 7. Let's see.If z = 0 is a root of multiplicity 7, then P(z)*S(z) - T(z) would be divisible by z^7. So, P(z)*S(z) = T(z) + z^7.But T(z) has a constant term of 20, and P(z)*S(z) has a constant term of 4*5 = 20. So, T(z) + z^7 would have a constant term of 20 as well, which matches P(z)*S(z)'s constant term.But what about the leading terms? P(z)*S(z) has leading term a*d*z^7, and T(z) has leading term h*z^7. So, P(z)*S(z) = T(z) + z^7 implies that a*d*z^7 = h*z^7 + z^7, so a*d = h + 1.Therefore, if I choose h = a*d - 1, then this would work. So, it's possible to have P(z)*S(z) - T(z) = z^7, which has a single root at z = 0 with multiplicity 7. But wait, z^7 is a polynomial with a single root at 0, but it's a root of multiplicity 7. So, in terms of distinct roots, it's just one distinct root.But earlier, I thought that was impossible because of the leading coefficient conflict, but actually, it's possible if we adjust the leading coefficients accordingly.Wait, so if I set T(z) = P(z)*S(z) - z^7, then T(z) would have leading coefficient a*d - 1. Since T(z) is given, its leading coefficient is fixed, so we can choose P(z) and S(z) such that a*d = h + 1, where h is the leading coefficient of T(z).Therefore, it is possible to have P(z)*S(z) - T(z) = z^7, which has a single distinct root at z = 0 with multiplicity 7. So, in this case, the equation P(z)*S(z) = T(z) would have only one distinct complex solution, z = 0.But wait, is z = 0 a solution? Let's check.If z = 0, then P(0) = 4, S(0) = 5, so P(0)*S(0) = 20, and T(0) = 20. So, yes, z = 0 is a solution.But if P(z)*S(z) - T(z) = z^7, then z = 0 is the only solution, with multiplicity 7. So, in terms of distinct solutions, it's just one.But earlier, I thought that was impossible because of the leading coefficient conflict, but it seems it's possible by adjusting the leading coefficients appropriately.Wait, but in the problem statement, T(z) is given with a constant term of 20, but its leading coefficient is not specified. So, as long as we can choose P(z) and S(z) such that a*d = h + 1, where h is the leading coefficient of T(z), then it's possible.Therefore, in this case, the minimal number of distinct complex solutions is 1.But wait, let me double-check. If P(z)*S(z) - T(z) = z^7, then T(z) = P(z)*S(z) - z^7. So, T(z) has leading term a*d*z^7 - z^7, which is (a*d - 1)z^7. So, the leading coefficient of T(z) is a*d - 1.But in the problem statement, T(z) is given as a polynomial with real coefficients, degree 7, and constant term 20. The leading coefficient is not specified, so we can choose it to be a*d - 1, which is possible because we can choose a and d freely as long as a*d - 1 equals the leading coefficient of T(z).Therefore, it's possible to have P(z)*S(z) - T(z) = z^7, which has only one distinct complex solution, z = 0.But wait, is z = 0 the only solution? Because z^7 = 0 only when z = 0, right? So, yes, z = 0 is the only solution, but it's a root of multiplicity 7.Therefore, the minimal number of distinct complex solutions is 1.But let me think again. If P(z)*S(z) - T(z) = z^7, then the equation P(z)*S(z) = T(z) is equivalent to z^7 = 0, which only has z = 0 as a solution, but with multiplicity 7. So, in terms of distinct solutions, it's just one.But wait, in the problem statement, it says "distinct complex solutions". So, does that mean that even though z = 0 is a root of multiplicity 7, it's still just one distinct solution? Yes, I think so.Therefore, the minimal number of distinct complex solutions is 1.But wait, let me think about another possibility. Suppose P(z)*S(z) - T(z) has two distinct roots, one with multiplicity 6 and another with multiplicity 1. Then, the number of distinct solutions would be 2.But is that possible? Let's see.Suppose P(z)*S(z) - T(z) = (z - a)^6*(z - b), where a and b are distinct complex numbers. Then, it would have two distinct roots, a and b.But can such a polynomial exist given the constraints on P(z), S(z), and T(z)?Well, let's see. If I choose a and b such that a is a root of multiplicity 6 and b is a simple root, then the polynomial would have two distinct roots.But how would that affect the leading coefficients?The leading term would be (z)^6*(z) = z^7, so the leading coefficient would be 1. But in our case, the leading coefficient of P(z)*S(z) - T(z) is a*d - h, which we can set to 1 by choosing a*d = h + 1.So, it's possible to have such a polynomial.But then, in this case, the equation P(z)*S(z) = T(z) would have two distinct complex solutions: a and b.But earlier, I thought that it's possible to have only one distinct solution. So, which one is it?Wait, the problem is asking for the minimal number of distinct complex solutions. So, if it's possible to have only one distinct solution, then that would be the minimal number.But earlier, I thought that having only one distinct solution is possible by setting P(z)*S(z) - T(z) = z^7, which has only z = 0 as a solution.But wait, in that case, z = 0 is a solution, but are there any other solutions? No, because z^7 = 0 only when z = 0.Therefore, in that case, the equation P(z)*S(z) = T(z) has only one distinct complex solution, z = 0.But wait, let me check if that's possible. If P(z)*S(z) - T(z) = z^7, then T(z) = P(z)*S(z) - z^7.So, T(z) is a polynomial of degree 7 with leading term (a*d - 1)z^7 and constant term 20.But P(z)*S(z) has constant term 4*5 = 20, and z^7 has constant term 0, so T(z) has constant term 20 - 0 = 20, which matches.Therefore, it's possible to have T(z) = P(z)*S(z) - z^7, which would make P(z)*S(z) - T(z) = z^7, which has only one distinct complex solution, z = 0.Therefore, the minimal number of distinct complex solutions is 1.But wait, let me think again. If P(z)*S(z) - T(z) = z^7, then the equation P(z)*S(z) = T(z) is equivalent to z^7 = 0, which only has z = 0 as a solution. So, yes, only one distinct solution.But in the problem statement, it says "distinct complex solutions". So, does that include multiplicities? No, it's about distinct solutions, so z = 0 is the only one.Therefore, the minimal number is 1.But wait, let me think about another angle. Suppose P(z)*S(z) - T(z) is a polynomial of degree 7 with real coefficients. Then, any non-real roots must come in conjugate pairs. So, the number of non-real roots must be even. Therefore, the number of distinct complex roots must be even, unless there are real roots.But in the case where P(z)*S(z) - T(z) = z^7, all roots are real (z = 0) with multiplicity 7. So, in that case, the number of distinct complex roots is 1, which is odd, but that's because all roots are real.Wait, but complex roots include real roots, so in that case, the number of distinct complex roots is 1, which is allowed because it's a real root.But if I have non-real roots, they have to come in pairs, so the number of distinct non-real roots must be even. But real roots can be odd in number.Therefore, the minimal number of distinct complex roots is 1, achieved when all roots are real and coincide at z = 0.Therefore, the answer is 1.But wait, let me check if there's a way to have zero solutions. That is, if P(z)*S(z) - T(z) is a non-zero polynomial with no roots. But that's impossible because every non-constant polynomial has at least one root in the complex plane. So, the minimal number of solutions is at least 1.Therefore, the minimal number of distinct complex solutions is 1.So, the answer is B) 1.

If 4xz + yw = some number and xw + yz = 6, the value of the expression ( 2x + y ) ( 2z + w ) is 15. What is the result of the first equation?

💡Okay, so I have this problem here: If 4xz + yw = some number and xw + yz = 6, the value of the expression (2x + y)(2z + w) is 15. What is the result of the first equation?Alright, let's break this down step by step. First, I need to understand what each part of the problem is saying.We have two equations:1) 4xz + yw = N (where N is the "some number" we need to find)2) xw + yz = 6And we're also given that:(2x + y)(2z + w) = 15So, the goal is to find the value of N in the first equation.Let me think about how these equations relate to each other. Maybe if I can express the third equation in terms of the first two, I can solve for N.First, let's expand the expression (2x + y)(2z + w). I'll use the distributive property (FOIL method) to do this.(2x + y)(2z + w) = 2x * 2z + 2x * w + y * 2z + y * wCalculating each term:2x * 2z = 4xz2x * w = 2xwy * 2z = 2yzy * w = ywSo, putting it all together:(2x + y)(2z + w) = 4xz + 2xw + 2yz + ywWe know from the problem that this equals 15:4xz + 2xw + 2yz + yw = 15Now, looking back at the second equation, we have xw + yz = 6. Maybe I can use this to substitute for some terms in the expanded expression.Notice that in the expanded expression, we have 2xw + 2yz. That's just 2 times (xw + yz). Since xw + yz = 6, then 2(xw + yz) = 2*6 = 12.So, substituting back into the expanded expression:4xz + 12 + yw = 15Now, let's simplify this:4xz + yw + 12 = 15Subtract 12 from both sides to isolate 4xz + yw:4xz + yw = 15 - 124xz + yw = 3But wait, 4xz + yw is exactly the left side of our first equation, which is equal to N. So, N = 3.Therefore, the result of the first equation is 3.I think that makes sense. Let me double-check my steps to make sure I didn't make any mistakes.1. Expanded (2x + y)(2z + w) correctly to get 4xz + 2xw + 2yz + yw.2. Recognized that 2xw + 2yz is 2*(xw + yz), which equals 12 since xw + yz = 6.3. Substituted back into the expanded expression to get 4xz + yw + 12 = 15.4. Subtracted 12 from both sides to find that 4xz + yw = 3.5. Concluded that N = 3.Everything seems to check out. I don't see any errors in my calculations or reasoning. So, I'm confident that the result of the first equation is indeed 3.

In the Cartesian coordinate system (xOy), let line y=-x+2 intersect circle x^{2}+y^{2}=r^{2} at points A and B, with O as the coordinate origin. If a point C on the circle satisfies overrightarrow{OC}=frac{5}{4}overrightarrow{OA}+frac{3}{4}overrightarrow{OB}, then r= _____.

💡Alright, so I have this problem here about a line intersecting a circle, and there's some vector stuff involved. Let me try to unpack it step by step.First, the problem says: In the Cartesian coordinate system (xOy), the line y = -x + 2 intersects the circle x² + y² = r² at points A and B. O is the origin. There's a point C on the circle such that the vector OC is equal to (5/4)OA + (3/4)OB. We need to find the radius r.Okay, so let me visualize this. There's a circle centered at the origin with radius r, and a line y = -x + 2 that cuts through the circle at points A and B. Then, there's another point C on the circle, and its position vector is a combination of vectors OA and OB. Interesting.I think I should start by finding the points of intersection between the line and the circle. That should give me points A and B. Once I have those, maybe I can figure out something about vectors OA and OB, and then use the given vector equation for OC to find r.So, let's find the intersection points of the line y = -x + 2 and the circle x² + y² = r². To do that, I can substitute y from the line equation into the circle equation.Substituting y = -x + 2 into x² + y² = r² gives:x² + (-x + 2)² = r²Let me expand that:x² + (x² - 4x + 4) = r²Combine like terms:x² + x² - 4x + 4 = r²So, 2x² - 4x + 4 = r²Let me write that as:2x² - 4x + (4 - r²) = 0This is a quadratic equation in terms of x. Let me denote it as:2x² - 4x + (4 - r²) = 0I can simplify this by dividing all terms by 2:x² - 2x + (2 - r²/2) = 0Hmm, not sure if that helps. Maybe I should keep it as 2x² - 4x + (4 - r²) = 0 for now.The solutions to this quadratic will give me the x-coordinates of points A and B. Let me denote the solutions as x₁ and x₂. Then, the corresponding y-coordinates will be y₁ = -x₁ + 2 and y₂ = -x₂ + 2.I know that for a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a.So, for our equation 2x² - 4x + (4 - r²) = 0:Sum of roots, x₁ + x₂ = 4/2 = 2Product of roots, x₁x₂ = (4 - r²)/2That might be useful later.Now, since points A and B lie on both the line and the circle, their coordinates satisfy both equations. So, OA and OB are vectors from the origin to these points, which means OA = (x₁, y₁) and OB = (x₂, y₂).Given that OC = (5/4)OA + (3/4)OB, and C is also on the circle, so |OC| = r.So, let's write that vector equation:OC = (5/4)OA + (3/4)OBWhich means:OC = (5/4)(x₁, y₁) + (3/4)(x₂, y₂)So, OC = ( (5/4)x₁ + (3/4)x₂ , (5/4)y₁ + (3/4)y₂ )Since C is on the circle, the magnitude of OC must be r. So,|OC|² = [ (5/4 x₁ + 3/4 x₂ )² + (5/4 y₁ + 3/4 y₂ )² ] = r²Let me compute that.First, let me compute the x-component squared:(5/4 x₁ + 3/4 x₂ )² = (25/16)x₁² + (9/16)x₂² + (2 * 5/4 * 3/4)x₁x₂Similarly, the y-component squared:(5/4 y₁ + 3/4 y₂ )² = (25/16)y₁² + (9/16)y₂² + (2 * 5/4 * 3/4)y₁y₂So, adding both components:25/16(x₁² + y₁²) + 9/16(x₂² + y₂²) + (15/8)(x₁x₂ + y₁y₂) = r²But since points A and B lie on the circle, x₁² + y₁² = r² and x₂² + y₂² = r². So, substituting:25/16(r²) + 9/16(r²) + (15/8)(x₁x₂ + y₁y₂) = r²Let me compute 25/16 + 9/16:25/16 + 9/16 = 34/16 = 17/8So, 17/8 r² + (15/8)(x₁x₂ + y₁y₂) = r²Subtract 17/8 r² from both sides:(15/8)(x₁x₂ + y₁y₂) = r² - 17/8 r²Which is:(15/8)(x₁x₂ + y₁y₂) = (8/8 - 17/8) r² = (-9/8) r²Multiply both sides by 8/15:x₁x₂ + y₁y₂ = (-9/8) r² * (8/15) = (-9/15) r² = (-3/5) r²So, x₁x₂ + y₁y₂ = (-3/5) r²Hmm, interesting. So, the dot product of vectors OA and OB is equal to (-3/5) r².But wait, the dot product of OA and OB is also equal to |OA||OB|cosθ, where θ is the angle between OA and OB. Since |OA| = |OB| = r, this becomes:OA · OB = r² cosθ = (-3/5) r²Therefore, cosθ = -3/5So, the angle θ between OA and OB has cosine equal to -3/5.Alright, so that's a useful piece of information.Now, I need to relate this to the line y = -x + 2 and the circle.Earlier, I found that the x-coordinates of A and B satisfy 2x² - 4x + (4 - r²) = 0, with sum of roots x₁ + x₂ = 2 and product x₁x₂ = (4 - r²)/2.Similarly, since y = -x + 2, the y-coordinates are y₁ = -x₁ + 2 and y₂ = -x₂ + 2.So, let's compute x₁x₂ + y₁y₂.We have:x₁x₂ + y₁y₂ = x₁x₂ + (-x₁ + 2)(-x₂ + 2)Let me expand that:= x₁x₂ + (x₁x₂ - 2x₁ - 2x₂ + 4)= x₁x₂ + x₁x₂ - 2x₁ - 2x₂ + 4= 2x₁x₂ - 2(x₁ + x₂) + 4We already know x₁ + x₂ = 2 and x₁x₂ = (4 - r²)/2.So, substituting:= 2*( (4 - r²)/2 ) - 2*2 + 4Simplify:= (4 - r²) - 4 + 4= (4 - r²) - 4 + 4= 4 - r²Wait, that's interesting. So, x₁x₂ + y₁y₂ = 4 - r²But earlier, we found that x₁x₂ + y₁y₂ = (-3/5) r²So, 4 - r² = (-3/5) r²Let me write that equation:4 - r² = (-3/5) r²Bring all terms to one side:4 = r² - (3/5) r²Simplify the right side:r² - (3/5) r² = (2/5) r²So, 4 = (2/5) r²Multiply both sides by 5/2:r² = 4 * (5/2) = 10Therefore, r = sqrt(10)Wait, that seems straightforward. Let me just verify my steps to make sure I didn't make a mistake.First, I found the intersection points by substituting y = -x + 2 into the circle equation, leading to a quadratic in x. Then, I used Vieta's formulas to find the sum and product of the roots.Then, I expressed OC as a linear combination of OA and OB, and since |OC| = r, I expanded the magnitude squared, substituted the known values for |OA| and |OB|, and ended up with an equation involving the dot product of OA and OB.I related the dot product to the cosine of the angle between OA and OB, which gave me cosθ = -3/5.Then, I computed x₁x₂ + y₁y₂ in two different ways: once using the coordinates and once using the dot product, and set them equal to each other.That led me to an equation where I could solve for r², which turned out to be 10, so r is sqrt(10).I think that makes sense. Let me just double-check the computation where I expanded x₁x₂ + y₁y₂.So, y₁ = -x₁ + 2 and y₂ = -x₂ + 2.Therefore, y₁y₂ = (-x₁ + 2)(-x₂ + 2) = x₁x₂ - 2x₁ - 2x₂ + 4.Then, x₁x₂ + y₁y₂ = x₁x₂ + x₁x₂ - 2x₁ - 2x₂ + 4 = 2x₁x₂ - 2(x₁ + x₂) + 4.Yes, that's correct.We had x₁ + x₂ = 2 and x₁x₂ = (4 - r²)/2.Substituting:2*( (4 - r²)/2 ) - 2*2 + 4 = (4 - r²) - 4 + 4 = 4 - r².Yes, that seems right.And we had from the vector equation that x₁x₂ + y₁y₂ = (-3/5) r².Therefore, 4 - r² = (-3/5) r².Solving for r²:4 = r² - (3/5) r² = (2/5) r²So, r² = 4 * (5/2) = 10.Therefore, r = sqrt(10).Yep, that seems consistent. I don't see any mistakes in the reasoning.**Final Answer**The radius ( r ) is (boxed{sqrt{10}}).

Six middle schools participated in a summer project:Six students from Allen school worked for 2 days.Five students from Balboa school worked for 6 days.Seven students from Carver school worked for 8 days.Three students from Dawson school worked for 4 days.Eight students from Elmwood school worked for 3 days.Four students from Franklin school worked for 7 days.The total amount paid for the students' work was 1494. Assuming each student received the same amount for a day's work, how much did the students from Elmwood school earn altogether?(A) 200.04 text{ dollars} (B) 221.33 text{ dollars} (C) 246.50 text{ dollars} (D) 259.22 text{ dollars}

💡To determine how much the students from Elmwood school earned altogether, we need to follow these steps:1. **Calculate the total number of student-days worked by all schools:** - Allen school: (6 text{ students} times 2 text{ days} = 12 text{ student-days}) - Balboa school: (5 text{ students} times 6 text{ days} = 30 text{ student-days}) - Carver school: (7 text{ students} times 8 text{ days} = 56 text{ student-days}) - Dawson school: (3 text{ students} times 4 text{ days} = 12 text{ student-days}) - Elmwood school: (8 text{ students} times 3 text{ days} = 24 text{ student-days}) - Franklin school: (4 text{ students} times 7 text{ days} = 28 text{ student-days}) Adding these together: [ 12 + 30 + 56 + 12 + 24 + 28 = 162 text{ student-days} ]2. **Determine the daily wage per student:** The total amount paid for all student-days is 1494. To find the daily wage per student, divide the total amount by the total number of student-days: [ text{Daily wage per student} = frac{1494}{162} approx 9.22222222 text{ dollars per day} ]3. **Calculate the total earnings for students from Elmwood school:** Elmwood school contributed (24) student-days. Multiply the daily wage by the number of student-days worked by Elmwood school: [ text{Total earnings for Elmwood school} = 9.22222222 text{ dollars/day} times 24 text{ student-days} = 221.3333333 text{ dollars} ]4. **Round the result to two decimal places:** [ 221.3333333 approx 221.33 text{ dollars} ]Thus, the students from Elmwood school earned altogether (boxed{221.33 text{ dollars}}).

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