💡Okay, so I have this problem about a hyperbola, and I need to find the value of |PF₁|·|PF₂| where P is a point on the hyperbola such that the vectors PF₁ and PF₂ are perpendicular. Hmm, let me try to break this down step by step.First, the equation of the hyperbola is given as x²/4 - y² = 1. I remember that the standard form of a hyperbola is x²/a² - y²/b² = 1, so in this case, a² is 4 and b² is 1. That means a is 2 and b is 1. Now, for hyperbolas, the distance to the foci from the center is given by c, where c² = a² + b². So let me calculate c. c² = a² + b² = 4 + 1 = 5, so c is sqrt(5). Therefore, the foci F₁ and F₂ are located at (-sqrt(5), 0) and (sqrt(5), 0) respectively. Alright, so F₁ is (-sqrt(5), 0) and F₂ is (sqrt(5), 0). The distance between F₁ and F₂ is 2c, which is 2*sqrt(5). Now, the point P is on the hyperbola, and the vectors PF₁ and PF₂ are perpendicular. That means their dot product is zero. So, if I denote the coordinates of P as (x, y), then the vectors PF₁ and PF₂ can be written as:PF₁ = (x - (-sqrt(5)), y - 0) = (x + sqrt(5), y)PF₂ = (x - sqrt(5), y - 0) = (x - sqrt(5), y)The dot product of PF₁ and PF₂ is:(x + sqrt(5))(x - sqrt(5)) + y * y = 0Let me compute that:(x + sqrt(5))(x - sqrt(5)) = x² - (sqrt(5))² = x² - 5y * y = y²So, the dot product is x² - 5 + y² = 0. Therefore, x² + y² = 5.But wait, P is on the hyperbola, so it must satisfy the hyperbola equation x²/4 - y² = 1. Let me write that down:x²/4 - y² = 1So, I have two equations:1. x² + y² = 5 (from the dot product condition)2. x²/4 - y² = 1 (from the hyperbola equation)I can solve these two equations simultaneously to find x² and y². Let me denote equation 2 as:x²/4 - y² = 1I can rearrange this to express y² in terms of x²:y² = x²/4 - 1Now, substitute this into equation 1:x² + (x²/4 - 1) = 5Simplify:x² + x²/4 - 1 = 5Combine like terms:(4x²/4 + x²/4) - 1 = 5(5x²/4) - 1 = 5Add 1 to both sides:5x²/4 = 6Multiply both sides by 4:5x² = 24Divide both sides by 5:x² = 24/5So, x² is 24/5. Now, let's find y² using equation 2:y² = x²/4 - 1 = (24/5)/4 - 1 = (6/5) - 1 = (6/5 - 5/5) = 1/5So, y² is 1/5. Therefore, y is either sqrt(1/5) or -sqrt(1/5), but since we're dealing with distances, the sign doesn't matter.Now, I need to find |PF₁| and |PF₂|. Let's compute these distances.First, |PF₁| is the distance from P(x, y) to F₁(-sqrt(5), 0):|PF₁| = sqrt[(x + sqrt(5))² + (y - 0)²] = sqrt[(x + sqrt(5))² + y²]Similarly, |PF₂| is the distance from P(x, y) to F₂(sqrt(5), 0):|PF₂| = sqrt[(x - sqrt(5))² + y²]I need to compute |PF₁|·|PF₂|. Let me denote this product as:|PF₁|·|PF₂| = sqrt[(x + sqrt(5))² + y²] * sqrt[(x - sqrt(5))² + y²]Hmm, this looks a bit complicated. Maybe I can square the product to simplify it:(|PF₁|·|PF₂|)² = [(x + sqrt(5))² + y²] * [(x - sqrt(5))² + y²]Let me compute each term inside the square roots first.Compute (x + sqrt(5))² + y²:= x² + 2x*sqrt(5) + 5 + y²Similarly, (x - sqrt(5))² + y²:= x² - 2x*sqrt(5) + 5 + y²So, the product becomes:[(x² + 2x*sqrt(5) + 5 + y²) * (x² - 2x*sqrt(5) + 5 + y²)]This looks like a difference of squares. Let me denote A = x² + y² + 5 and B = 2x*sqrt(5). Then, the product is (A + B)(A - B) = A² - B².So, let's compute A² - B²:A² = (x² + y² + 5)²B² = (2x*sqrt(5))² = 4x²*5 = 20x²So, the product is:(x² + y² + 5)² - 20x²But from equation 1, we know that x² + y² = 5. So, substitute that into A:A = 5 + 5 = 10Therefore, A² = 10² = 100So, the product becomes:100 - 20x²But we already found that x² = 24/5, so:20x² = 20*(24/5) = 4*24 = 96Therefore, the product is:100 - 96 = 4So, (|PF₁|·|PF₂|)² = 4, which means |PF₁|·|PF₂| = sqrt(4) = 2Wait, but let me double-check my steps because I might have made a mistake.First, when I squared the product, I got (|PF₁|·|PF₂|)² = 4, so taking the square root gives 2. That seems correct.Alternatively, maybe there's a smarter way to compute |PF₁|·|PF₂| without expanding everything.I remember that for hyperbolas, the difference of distances from any point on the hyperbola to the foci is constant, which is 2a. In this case, 2a = 4. So, |PF₁| - |PF₂| = 4 or |PF₂| - |PF₁| = 4, depending on which branch P is on.But in this case, since the dot product is zero, the triangle PF₁F₂ is right-angled at P. So, by the Pythagorean theorem:|PF₁|² + |PF₂|² = |F₁F₂|²We know |F₁F₂| is 2c = 2*sqrt(5), so |F₁F₂|² = (2*sqrt(5))² = 20.So, |PF₁|² + |PF₂|² = 20.Also, from the hyperbola definition, | |PF₁| - |PF₂| | = 2a = 4.Let me denote |PF₁| = d and |PF₂| = e.So, we have:d² + e² = 20and|d - e| = 4We need to find d*e.Let me square the second equation:(d - e)² = 16Which gives:d² - 2de + e² = 16But we know d² + e² = 20, so substitute:20 - 2de = 16Therefore:-2de = 16 - 20 = -4Divide both sides by -2:de = 2So, d*e = 2, which is |PF₁|·|PF₂| = 2.That's the same result as before, so that confirms it.Therefore, the value is 2, which is option A.**Final Answer**boxed{A}

💡Okay, so I have this problem here: Let ( b_n = frac{8^n - 1}{7} ). Define ( e_n ) to be the greatest common divisor (gcd) of ( b_n ) and ( b_{n+1} ). I need to determine the maximum possible value that ( e_n ) can achieve. Hmm, interesting. Let me try to figure this out step by step.First, let me write down what ( b_n ) and ( b_{n+1} ) are. So, ( b_n = frac{8^n - 1}{7} ) and ( b_{n+1} = frac{8^{n+1} - 1}{7} ). I need to find the gcd of these two expressions. I remember that the Euclidean algorithm is useful for finding the gcd of two numbers. Maybe I can apply that here. The Euclidean algorithm says that ( gcd(a, b) = gcd(b, a mod b) ). So, perhaps I can express ( b_{n+1} ) in terms of ( b_n ) and then apply the algorithm.Let me compute ( b_{n+1} ) in terms of ( b_n ). So, ( b_{n+1} = frac{8^{n+1} - 1}{7} = frac{8 cdot 8^n - 1}{7} ). Hmm, I can rewrite this as ( frac{8(8^n - 1) + 8 - 1}{7} ). Let's see, that becomes ( frac{8(8^n - 1) + 7}{7} ). Breaking that down, ( frac{8(8^n - 1)}{7} + frac{7}{7} ) which simplifies to ( 8 cdot frac{8^n - 1}{7} + 1 ). But ( frac{8^n - 1}{7} ) is exactly ( b_n ). So, ( b_{n+1} = 8b_n + 1 ). Alright, so now I have ( b_{n+1} = 8b_n + 1 ). Let me plug this into the Euclidean algorithm. The gcd of ( b_n ) and ( b_{n+1} ) is the same as the gcd of ( b_n ) and ( b_{n+1} - 8b_n ). Because in the Euclidean algorithm, subtracting a multiple of one number from the other doesn't change the gcd.So, ( gcd(b_n, b_{n+1}) = gcd(b_n, b_{n+1} - 8b_n) ). Let's compute ( b_{n+1} - 8b_n ). From above, ( b_{n+1} - 8b_n = (8b_n + 1) - 8b_n = 1 ). Therefore, ( gcd(b_n, b_{n+1}) = gcd(b_n, 1) ). But the gcd of any number and 1 is always 1, since 1 divides every integer. So, this simplifies to ( gcd(b_n, 1) = 1 ).Wait, so does that mean that ( e_n = 1 ) for all ( n )? That seems surprising. Let me test this with some small values of ( n ) to make sure I didn't make a mistake.Let's take ( n = 1 ). Then, ( b_1 = frac{8^1 - 1}{7} = frac{7}{7} = 1 ). ( b_2 = frac{8^2 - 1}{7} = frac{64 - 1}{7} = frac{63}{7} = 9 ). The gcd of 1 and 9 is indeed 1.Now, ( n = 2 ). ( b_2 = 9 ) as above, ( b_3 = frac{8^3 - 1}{7} = frac{512 - 1}{7} = frac{511}{7} = 73 ). The gcd of 9 and 73 is 1 because 73 is a prime number and doesn't divide 9.How about ( n = 3 )? ( b_3 = 73 ), ( b_4 = frac{8^4 - 1}{7} = frac{4096 - 1}{7} = frac{4095}{7} = 585 ). The gcd of 73 and 585. Let's see, 585 divided by 73 is about 8 times, 73*8=584, so 585 - 584 =1. So, the gcd is 1.Hmm, seems consistent. Let me try ( n = 4 ). ( b_4 = 585 ), ( b_5 = frac{8^5 - 1}{7} = frac{32768 - 1}{7} = frac{32767}{7} = 4681 ). The gcd of 585 and 4681. Let's compute 4681 divided by 585. 585*8=4680, so 4681 - 4680=1. So, again, the gcd is 1.Wait, so in all these cases, the gcd is 1. So, does that mean that ( e_n = 1 ) always? That seems to be the case.But just to be thorough, let me consider another approach. Maybe using properties of exponents or modular arithmetic.I know that ( 8 equiv 1 mod 7 ), since 8 - 1 = 7, which is divisible by 7. So, ( 8^n equiv 1^n equiv 1 mod 7 ). Therefore, ( 8^n - 1 equiv 0 mod 7 ), which is why ( b_n ) is an integer.But how does this help with the gcd? Maybe looking at the expression modulo ( e_n ). Since ( e_n ) divides both ( b_n ) and ( b_{n+1} ), it must divide any linear combination of them.In particular, ( e_n ) divides ( b_{n+1} - 8b_n ), which we already found to be 1. Therefore, ( e_n ) divides 1, which means ( e_n = 1 ).So, regardless of the value of ( n ), the gcd ( e_n ) is always 1. Therefore, the maximum possible value that ( e_n ) can achieve is 1.I think that's solid. I've checked it with specific examples and also through algebraic manipulation and properties of modular arithmetic. It consistently shows that the gcd is 1. So, yeah, I'm confident that the maximum value of ( e_n ) is 1.**Final Answer**The maximum possible value of ( e_n ) is boxed{1}.

💡First, I need to determine the principal amount ( x ) based on the given information about simple interest and compound interest.1. **Simple Interest Calculation:** - The formula for simple interest is ( SI = frac{P times r times t}{100} ). - Given that the simple interest earned over two years is 600, I can set up the equation: [ 600 = frac{x times y times 2}{100} ] - Simplifying this equation gives: [ x times y = 30,000 quad text{(Equation 1)} ]2. **Compound Interest Calculation:** - The formula for compound interest is ( CI = P times left(1 + frac{r}{100}right)^t - P ). - Given that the compound interest earned over two years is 615, I can set up the equation: [ 615 = x times left(1 + frac{y}{100}right)^2 - x ] - Simplifying this equation gives: [ 615 = x times left(frac{y}{100} + frac{y^2}{10,000}right) ] [ 61,500 = x times y + frac{x times y^2}{100} quad text{(Equation 2)} ]3. **Solving the Equations:** - From Equation 1, I know that ( x times y = 30,000 ). - Substituting ( x times y ) into Equation 2: [ 61,500 = 30,000 + frac{x times y^2}{100} ] [ 31,500 = frac{x times y^2}{100} ] [ x times y^2 = 3,150,000 quad text{(Equation 3)} ] - Dividing Equation 3 by Equation 1 to eliminate ( x ): [ frac{x times y^2}{x times y} = frac{3,150,000}{30,000} ] [ y = 105 ] - Substituting ( y = 105 ) back into Equation 1: [ x times 105 = 30,000 ] [ x = frac{30,000}{105} approx 285.71 ]Therefore, the value of ( x ) is approximately 285.71.

💡Okay, so I've got this problem about an arithmetic sequence. Let me try to figure it out step by step. The problem says:"Given an arithmetic sequence {a_n} with the sum of the first n terms as S_n, a_6 < 0, and a_4 + a_9 > 0, the largest value of n that satisfies the inequality S_n < 0 is ( )."The options are A:9, B:10, C:11, D:12.First, I need to recall what an arithmetic sequence is. It's a sequence where each term increases by a constant difference, called the common difference, usually denoted as d. So, the nth term can be written as a_n = a_1 + (n-1)d, where a_1 is the first term.The sum of the first n terms, S_n, is given by the formula S_n = n/2 * (2a_1 + (n-1)d) or sometimes written as S_n = n*(a_1 + a_n)/2. Both formulas are equivalent.Now, the problem gives me two conditions:1. a_6 < 02. a_4 + a_9 > 0I need to find the largest n such that S_n < 0.Let me try to translate these conditions into something I can work with.First, let's write expressions for a_6, a_4, and a_9.a_6 = a_1 + 5d < 0a_4 = a_1 + 3da_9 = a_1 + 8dSo, the second condition is a_4 + a_9 > 0, which is:(a_1 + 3d) + (a_1 + 8d) > 0Simplify that:2a_1 + 11d > 0So, now I have two inequalities:1. a_1 + 5d < 02. 2a_1 + 11d > 0I can write these as:1. a_1 < -5d2. 2a_1 > -11d => a_1 > -11d/2So, combining these two inequalities:-11d/2 < a_1 < -5dHmm, that's interesting. So, a_1 is between -11d/2 and -5d.I wonder if I can find the sign of d. Let's see.If d is positive, then the sequence is increasing. If d is negative, the sequence is decreasing.Looking at a_6 < 0 and a_4 + a_9 > 0.If d is positive, then the terms are increasing. So, a_6 is less than 0, but a_9 is greater than a_6 because d is positive. Similarly, a_4 is less than a_5, which is less than a_6, etc.But a_4 + a_9 > 0. So, even though a_6 is negative, a_9 is positive enough to make the sum positive.Alternatively, if d is negative, the sequence is decreasing. So, a_6 < 0, and a_9 would be even more negative, but a_4 would be less negative than a_6. So, adding a_4 and a_9 would be adding two negative numbers, which might not necessarily be positive. So, maybe d is positive.Wait, let's think about that.If d is positive, then the sequence is increasing. So, a_6 is negative, but a_7, a_8, a_9, etc., are more positive. So, a_9 is positive, and a_4 is less than a_5, which is less than a_6, which is negative. So, a_4 is also negative, but a_9 is positive. So, a_4 + a_9 could be positive if a_9 is sufficiently positive.If d is negative, then the sequence is decreasing. So, a_6 is negative, and a_9 is even more negative, while a_4 is less negative than a_6. So, adding a_4 and a_9 would be adding two negative numbers, which would make it more negative, which contradicts the condition a_4 + a_9 > 0. Therefore, d must be positive.So, d > 0.Therefore, the sequence is increasing, starting from a_1, and eventually becoming positive.Given that, let's see.We have a_6 < 0, so the 6th term is negative.But a_4 + a_9 > 0.Since the sequence is increasing, a_9 is greater than a_6, which is negative, so a_9 could be positive.Similarly, a_4 is less than a_6, so a_4 is also negative, but maybe not as negative as a_6.So, adding a_4 and a_9, which are negative and positive respectively, gives a positive result.So, that makes sense.So, now, I need to find the largest n such that S_n < 0.Since the sequence is increasing, the sum S_n will initially be negative, reach a minimum, and then start increasing.So, the sum S_n will be negative up to a certain point, and then become positive.We need to find the largest n where S_n is still negative.So, perhaps S_n is negative up to n=11, and becomes positive at n=12.But let's verify that.First, let's express S_n in terms of a_1 and d.S_n = n/2 * (2a_1 + (n-1)d)Alternatively, S_n = n*(a_1 + a_n)/2Since a_n = a_1 + (n-1)d.So, S_n = n*(a_1 + a_1 + (n-1)d)/2 = n*(2a_1 + (n-1)d)/2So, S_n = n*(2a_1 + (n-1)d)/2We can write this as S_n = n*(a_1 + (n-1)d/2)But maybe it's better to keep it as S_n = n*(2a_1 + (n-1)d)/2Now, we have two inequalities from the given conditions:1. a_1 + 5d < 02. 2a_1 + 11d > 0So, let's write these as:1. a_1 < -5d2. 2a_1 > -11d => a_1 > -11d/2So, combining:-11d/2 < a_1 < -5dSo, a_1 is between -11d/2 and -5d.Since d is positive, as we established earlier.So, let's try to express S_n in terms of d.Let me substitute a_1 from the inequalities.From the first inequality, a_1 < -5dFrom the second inequality, a_1 > -11d/2So, let's pick a value for a_1 in this range. Maybe the midpoint or something, but perhaps it's better to express S_n in terms of d.Alternatively, maybe we can express S_n in terms of a_6 or something.Wait, let's think about the sum S_n.Since the sequence is arithmetic, the sum S_n is a quadratic function of n.So, S_n = An^2 + Bn, where A and B are constants.Given that, the graph of S_n vs n is a parabola opening upwards or downwards.Since d is positive, the sequence is increasing, so the terms go from negative to positive.Therefore, the sum S_n will first decrease, reach a minimum, and then increase.Therefore, the parabola opens upwards.So, the vertex of the parabola is the minimum point.We need to find the value of n where S_n is still negative, just before it becomes positive.So, the largest n such that S_n < 0.So, perhaps n is around 11 or 12.But let's calculate.Alternatively, maybe we can find the value of n where S_n = 0, and then take the floor of that.But perhaps it's better to find n such that S_n < 0.Alternatively, maybe we can find the term where a_n changes sign.Since the sequence is increasing, there exists some k such that a_k < 0 and a_{k+1} > 0.So, the terms before k are negative, and after k are positive.So, the sum S_n will be the sum of negative terms up to k, and then adding positive terms beyond that.So, the sum will decrease until n=k, and then start increasing.Therefore, the minimum sum occurs at n=k.But we need the largest n where S_n < 0.So, perhaps the sum becomes positive at some n > k.So, we need to find the point where the sum crosses zero.Alternatively, maybe we can find k such that a_k < 0 and a_{k+1} > 0.Given that a_6 < 0, and a_4 + a_9 > 0.We know that a_6 < 0, and a_9 > -a_4.But since a_4 is negative, a_9 is positive.Wait, let's see.From a_4 + a_9 > 0, and a_4 is negative, a_9 must be positive enough to make the sum positive.So, a_9 > -a_4.But a_4 = a_1 + 3d, and a_9 = a_1 + 8d.So, a_9 = a_4 + 5d.So, a_9 = a_4 + 5d > -a_4Therefore, a_4 + 5d > -a_4So, 2a_4 + 5d > 0But a_4 = a_1 + 3d, so:2(a_1 + 3d) + 5d > 02a_1 + 6d + 5d > 02a_1 + 11d > 0Which is the second condition given.So, that's consistent.Now, let's try to find the value of k where a_k < 0 and a_{k+1} > 0.So, a_k = a_1 + (k-1)d < 0a_{k+1} = a_1 + kd > 0So, we have:a_1 + (k-1)d < 0a_1 + kd > 0Subtracting the first inequality from the second:d > 0Which is consistent with d being positive.So, the term where the sequence crosses zero is between k and k+1.So, let's solve for k.From a_1 + (k-1)d < 0a_1 < -(k-1)dFrom a_1 + kd > 0a_1 > -kdSo, combining:-kd < a_1 < -(k-1)dBut we also have from the given conditions:-11d/2 < a_1 < -5dSo, we can set up:-11d/2 < a_1 < -5dand-kd < a_1 < -(k-1)dSo, we need to find k such that:-11d/2 < -(k-1)d and -kd < -5dWait, let's see.From the first set:a_1 > -11d/2 and a_1 < -5dFrom the second set:a_1 > -kd and a_1 < -(k-1)dSo, to have overlap, we need:-11d/2 < -(k-1)d and -kd < -5dLet's solve these inequalities.First inequality:-11d/2 < -(k-1)dDivide both sides by d (since d > 0, inequality sign remains):-11/2 < -(k-1)Multiply both sides by -1 (inequality sign reverses):11/2 > k - 1So,k - 1 < 11/2k < 13/2k < 6.5Since k is an integer, k ≤ 6Second inequality:-kd < -5dDivide both sides by d (positive, so inequality remains):-k < -5Multiply both sides by -1 (reverse inequality):k > 5So, k > 5So, combining both inequalities:5 < k ≤ 6.5Since k is an integer, k = 6So, the term a_6 < 0 and a_7 > 0Therefore, the sequence changes sign between term 6 and term 7.So, up to term 6, all terms are negative, and from term 7 onwards, terms are positive.Now, the sum S_n will be the sum of negative terms up to n=6, and then adding positive terms beyond that.So, the sum S_n will decrease until n=6, reach a minimum, and then start increasing.But wait, actually, the sum S_n is a quadratic function, so it will have a minimum at some point.But since the terms are negative up to n=6, and positive beyond, the sum will decrease until n=6, then start increasing.But the sum could still be negative even after n=6, because the positive terms added might not yet overcome the negative sum.So, we need to find the largest n where S_n < 0.So, let's try to find when S_n becomes positive.Alternatively, maybe we can find the value of n where S_n = 0, and then take the floor of that.But let's see.We can write S_n = n/2 [2a_1 + (n-1)d]We need to find n such that S_n < 0.Given that a_1 is between -11d/2 and -5d.Let me express a_1 in terms of d.From the inequalities:-11d/2 < a_1 < -5dLet me pick a value for a_1 in this range to make calculations easier.Let's say a_1 = -6d.Is that within the range?-11d/2 = -5.5d, so -6d is less than -5.5d, which is outside the range.Wait, so a_1 must be greater than -11d/2, which is -5.5d.So, let's pick a_1 = -5.25d, which is between -5.5d and -5d.But maybe it's better to keep it symbolic.Alternatively, let's express S_n in terms of d.Given that a_1 is between -11d/2 and -5d.So, let's write S_n = n/2 [2a_1 + (n-1)d]We can write this as:S_n = n/2 [2a_1 + dn - d]= n/2 [2a_1 - d + dn]= n/2 [2a_1 - d + dn]But maybe it's better to keep it as:S_n = n/2 [2a_1 + (n-1)d]We can substitute a_1 from the inequalities.From the first inequality, a_1 < -5dFrom the second inequality, a_1 > -11d/2So, let's substitute a_1 = -5d + ε, where ε is a small positive number.But maybe that's complicating.Alternatively, let's express S_n in terms of a_6.Since a_6 = a_1 + 5d < 0So, a_1 = a_6 - 5dSo, S_n = n/2 [2(a_6 - 5d) + (n-1)d]= n/2 [2a_6 - 10d + dn - d]= n/2 [2a_6 - 11d + dn]= n/2 [dn + 2a_6 - 11d]= n/2 [d(n - 11) + 2a_6]But since a_6 < 0, and d > 0, this might help.Alternatively, maybe we can find the value of n where S_n = 0.Set S_n = 0:n/2 [2a_1 + (n-1)d] = 0Since n ≠ 0, we have:2a_1 + (n-1)d = 0So,2a_1 = -(n-1)da_1 = -(n-1)d/2But from the given conditions, we have:-11d/2 < a_1 < -5dSo,-11d/2 < -(n-1)d/2 < -5dMultiply all parts by 2/d (since d > 0, inequality signs remain):-11 < -(n-1) < -10Multiply by -1 (reverse inequalities):11 > n - 1 > 10So,10 < n - 1 < 11Add 1:11 < n < 12Since n must be an integer, n = 11 is the only integer in this interval.Therefore, S_n = 0 when n = 11.Wait, that can't be, because n must be an integer, and S_n is a quadratic function, so it can cross zero at a non-integer n.Wait, let me double-check.We set S_n = 0:2a_1 + (n-1)d = 0So,n = 1 - 2a_1/dBut from the inequalities:-11d/2 < a_1 < -5dSo,-11d/2 < a_1 < -5dMultiply all parts by -2/d (since d > 0, inequality signs reverse):11 > 2a_1/d > 10So,10 < 2a_1/d < 11Therefore,n = 1 - 2a_1/dSo,n = 1 - (something between 10 and 11)So,n = 1 - 10.5 (for example) = -9.5Wait, that doesn't make sense.Wait, maybe I made a mistake in the algebra.Let me go back.We have:2a_1 + (n-1)d = 0So,n = 1 - 2a_1/dFrom the inequalities:-11d/2 < a_1 < -5dSo,-11d/2 < a_1 < -5dMultiply all parts by -2/d:11 > 2a_1/d > 10So,10 < 2a_1/d < 11Therefore,n = 1 - (something between 10 and 11)So,n = 1 - 10.5 = -9.5Wait, that can't be right because n is positive.Wait, perhaps I made a mistake in the direction of the inequality.Wait, when I multiply by a negative, the inequality signs reverse.So, from:-11d/2 < a_1 < -5dMultiply by -2/d:11 > 2a_1/d > 10So,10 < 2a_1/d < 11Therefore,n = 1 - 2a_1/dSo,n = 1 - (something between 10 and 11)So,n = 1 - 10.5 = -9.5Wait, that's negative, which doesn't make sense because n is positive.Hmm, maybe I messed up the substitution.Wait, let's think differently.We have:2a_1 + (n-1)d = 0So,n = 1 - 2a_1/dBut from the inequalities:-11d/2 < a_1 < -5dSo,-11d/2 < a_1 < -5dMultiply by -2/d (inequality signs reverse):11 > 2a_1/d > 10So,10 < 2a_1/d < 11Therefore,n = 1 - (something between 10 and 11)So,n = 1 - 10.5 = -9.5Wait, that's still negative.This suggests that n would be negative, which doesn't make sense.But we know that S_n is negative for some positive n.Wait, maybe I need to approach this differently.Let me consider that S_n is a quadratic function of n, opening upwards because d > 0.So, the vertex of the parabola is at n = -B/(2A), where S_n = An^2 + Bn.From S_n = n/2 [2a_1 + (n-1)d] = (2a_1 n + dn(n-1))/2 = (2a_1 n + dn^2 - dn)/2 = (dn^2 + (2a_1 - d)n)/2So, S_n = (d/2)n^2 + (a_1 - d/2)nSo, A = d/2, B = a_1 - d/2So, the vertex is at n = -B/(2A) = -(a_1 - d/2)/(2*(d/2)) = -(a_1 - d/2)/d = (-a_1 + d/2)/d = (-a_1)/d + 1/2So, n_vertex = (-a_1)/d + 1/2From the inequalities:-11d/2 < a_1 < -5dSo,-11d/2 < a_1 < -5dMultiply by -1/d (inequality signs reverse):11/2 > -a_1/d > 5So,5 < -a_1/d < 5.5Therefore,n_vertex = (-a_1)/d + 1/2So,5 + 0.5 < n_vertex < 5.5 + 0.5So,5.5 < n_vertex < 6So, the vertex is between n=5.5 and n=6.So, the minimum sum occurs around n=5.5 to n=6.So, the sum S_n is decreasing until n≈5.5-6, then starts increasing.So, the sum is negative up to some n, then becomes positive.We need to find the largest n where S_n < 0.So, let's compute S_n for n=11 and n=12.But first, let's see if we can express S_n in terms of a_6.We know that a_6 = a_1 + 5d < 0So, a_1 = a_6 - 5dSo, S_n = n/2 [2a_1 + (n-1)d] = n/2 [2(a_6 - 5d) + (n-1)d] = n/2 [2a_6 - 10d + dn - d] = n/2 [2a_6 - 11d + dn]= n/2 [d(n - 11) + 2a_6]Since a_6 < 0, and d > 0, let's see.For n=11:S_11 = 11/2 [d(11 - 11) + 2a_6] = 11/2 [0 + 2a_6] = 11a_6Since a_6 < 0, S_11 = 11a_6 < 0For n=12:S_12 = 12/2 [d(12 - 11) + 2a_6] = 6 [d + 2a_6]But we know that a_4 + a_9 > 0a_4 = a_1 + 3d, a_9 = a_1 + 8dSo, a_4 + a_9 = 2a_1 + 11d > 0But a_1 = a_6 - 5d, so:2(a_6 - 5d) + 11d > 02a_6 - 10d + 11d > 02a_6 + d > 0So, 2a_6 + d > 0Therefore, d > -2a_6But since a_6 < 0, -2a_6 is positive.So, d > -2a_6Therefore, in S_12 = 6(d + 2a_6)But from above, d + 2a_6 > 0Because d > -2a_6 => d + 2a_6 > 0Therefore, S_12 = 6(d + 2a_6) > 0So, S_12 > 0Therefore, S_11 < 0 and S_12 > 0So, the largest n where S_n < 0 is n=11Therefore, the answer is C:11

💡Okay, so I have this problem here: I need to find the equation of the parabola that's symmetric to the parabola y = 2(x - 3)^2 - 5 about the x-axis. Hmm, okay. Let me think about how to approach this.First, I remember that reflecting a graph over the x-axis involves changing the sign of the y-values. So, if I have a point (x, y) on the original parabola, its reflection over the x-axis would be (x, -y). That makes sense because reflecting over the x-axis flips the graph vertically.Looking at the given equation, y = 2(x - 3)^2 - 5, I can see that it's in the vertex form of a parabola, which is y = a(x - h)^2 + k. In this case, the vertex (h, k) is (3, -5). The coefficient 'a' is 2, which tells me that the parabola opens upwards because 'a' is positive.Now, if I want to reflect this parabola over the x-axis, I need to change the direction it opens. Since it was opening upwards before, after reflection, it should open downwards. To achieve this, I should change the sign of the coefficient 'a'. So, instead of 2, it should be -2.But wait, is that all? Let me think. The vertex was at (3, -5). If I reflect this over the x-axis, the x-coordinate remains the same, but the y-coordinate changes sign. So, the new vertex should be at (3, 5). That makes sense because reflecting over the x-axis flips the point vertically.So, putting that together, the new equation should have the same vertex form but with 'a' as -2 and the vertex at (3, 5). Therefore, the equation should be y = -2(x - 3)^2 + 5.Let me double-check that. If I take a point on the original parabola, say when x = 3, y = -5. Reflecting that over the x-axis would give me (3, 5), which is indeed the vertex of the new parabola. That seems correct.Just to be thorough, let me pick another point. Suppose x = 4. On the original parabola, y = 2(4 - 3)^2 - 5 = 2(1)^2 - 5 = 2 - 5 = -3. So, the point (4, -3) is on the original parabola. Reflecting this over the x-axis should give me (4, 3). Let's see if that point lies on the new parabola: y = -2(4 - 3)^2 + 5 = -2(1)^2 + 5 = -2 + 5 = 3. Perfect, it does.Let me try one more point. If x = 2, then on the original parabola, y = 2(2 - 3)^2 - 5 = 2(-1)^2 - 5 = 2(1) - 5 = 2 - 5 = -3. So, the point (2, -3) is on the original parabola. Reflecting over the x-axis should give me (2, 3). Plugging into the new equation: y = -2(2 - 3)^2 + 5 = -2(-1)^2 + 5 = -2(1) + 5 = -2 + 5 = 3. Yep, that works too.So, it seems like changing the sign of the coefficient and adjusting the vertex y-coordinate by changing its sign gives me the correct reflection over the x-axis. Therefore, the equation of the symmetric parabola should be y = -2(x - 3)^2 + 5.I think that's it. I don't see any mistakes in my reasoning. The key steps were recognizing the vertex form, understanding the effect of reflecting over the x-axis on the vertex and the direction the parabola opens, and then verifying with specific points to ensure the reflection is correct.

💡Alright, so I have this problem about a square with two lines drawn inside it, creating four triangles. I need to figure out what fraction of the square's area is shaded when two opposite triangles are colored. Hmm, okay, let me try to visualize this.First, the square. Let's say it's a standard square with sides of length 's'. The problem mentions midpoints of the sides, so I should probably consider the midpoints. If I imagine the square on a coordinate plane, with the bottom-left corner at (0,0) and the top-right corner at (s,s), then the midpoints would be at (s/2, 0) on the bottom side, (s, s/2) on the right side, (s/2, s) on the top side, and (0, s/2) on the left side.Now, the lines are drawn from the midpoint of the bottom side to the midpoint of the right side, and from the midpoint of the top side to the midpoint of the left side. So, one line goes from (s/2, 0) to (s, s/2), and the other goes from (s/2, s) to (0, s/2). These two lines intersect somewhere inside the square, creating four triangles.I think the key here is to figure out the areas of these triangles. Since the square is divided into four triangles by these two lines, each triangle should have the same area, right? But wait, I'm not sure if they are all equal. Let me think.If I draw these lines, they should intersect at the center of the square, which is at (s/2, s/2). So, each triangle has a vertex at the center. Now, each triangle is formed by connecting the center to the midpoints of the sides. So, each triangle has a base that's half the length of the square's side and a height that's also half the side length.Let me calculate the area of one triangle. The formula for the area of a triangle is (base * height) / 2. Here, the base is s/2 and the height is s/2. So, the area would be (s/2 * s/2) / 2 = (s²/4) / 2 = s²/8. So, each triangle has an area of s²/8.Since there are four such triangles, the total area covered by the triangles is 4 * (s²/8) = s²/2. Wait, but the area of the square is s², so that would mean half the square is covered by the triangles? That doesn't make sense because the lines are drawn inside the square, so the triangles should cover the entire area, right?Hmm, maybe I made a mistake in calculating the area. Let me check again. If each triangle has an area of s²/8, then four triangles would be 4*(s²/8) = s²/2. But the square's area is s², so that would imply that the triangles only cover half the square. That doesn't seem right because the lines are drawn from midpoints, so they should divide the square into four equal parts.Wait, perhaps the triangles are not all the same size? Maybe some are larger and some are smaller. Let me think about the geometry again.If I draw a line from the midpoint of the bottom side to the midpoint of the right side, that line will pass through the center of the square. Similarly, the line from the midpoint of the top side to the midpoint of the left side will also pass through the center. So, these two lines intersect at the center, dividing the square into four triangles.Each triangle is formed by two of these lines and a side of the square. Let me consider one of these triangles. For example, the triangle in the bottom-right corner. It has vertices at (s/2, 0), (s, s/2), and (s, 0). Wait, is that correct?Actually, no. The triangle in the bottom-right corner would have vertices at (s, 0), (s, s/2), and (s/2, s/2). Similarly, the triangle in the top-left corner would have vertices at (0, s), (0, s/2), and (s/2, s/2). The other two triangles would be in the top-right and bottom-left corners.So, each triangle is actually a right triangle with legs of length s/2. Therefore, the area of each triangle is (s/2 * s/2)/2 = s²/8. So, each triangle is s²/8, and there are four of them, totaling s²/2. Wait, that still doesn't add up because the square's area is s².I must be misunderstanding something. Maybe the triangles overlap or something? No, the lines are drawn from midpoints to midpoints, so they should divide the square into four distinct triangles without overlapping.Wait, perhaps the triangles are not all congruent? Maybe some are larger than others. Let me try to calculate the area of one triangle more carefully.Take the triangle in the bottom-right corner. Its vertices are at (s, 0), (s, s/2), and (s/2, s/2). So, the base is from (s, 0) to (s, s/2), which is a vertical line of length s/2. The height is from (s, s/2) to (s/2, s/2), which is a horizontal line of length s/2. So, the area is (base * height)/2 = (s/2 * s/2)/2 = s²/8.Similarly, the triangle in the top-left corner has vertices at (0, s), (0, s/2), and (s/2, s/2). Its base is from (0, s) to (0, s/2), length s/2, and height from (0, s/2) to (s/2, s/2), length s/2. So, area is also s²/8.Now, what about the other two triangles? The triangle in the top-right corner has vertices at (s, s), (s, s/2), and (s/2, s/2). Similarly, the triangle in the bottom-left corner has vertices at (0, 0), (0, s/2), and (s/2, s/2). Each of these also has an area of s²/8.So, all four triangles have the same area, s²/8, totaling s²/2. But the square's area is s², so where is the other half? I must be missing something.Wait, no, actually, the lines divide the square into four triangles and a central quadrilateral. So, the four triangles each have area s²/8, and the central quadrilateral has area s² - 4*(s²/8) = s² - s²/2 = s²/2. So, the central part is also s²/2.But the problem says that two opposite triangles are shaded. So, if each triangle is s²/8, then two triangles would be 2*(s²/8) = s²/4. Therefore, the shaded area is s²/4, and the total area is s², so the fraction is 1/4.Wait, but earlier I thought the four triangles only cover half the square, but actually, the four triangles plus the central quadrilateral make up the whole square. So, the four triangles are s²/2 in total, and the central quadrilateral is also s²/2.But the problem says that two opposite triangles are shaded. So, if each triangle is s²/8, then two of them would be s²/4, which is 1/4 of the square's area.So, the answer should be 1/4, which is option B.But let me double-check to make sure I didn't make a mistake. If I consider the square divided into four triangles by the two lines, each triangle has area s²/8, so two triangles would be s²/4. That seems correct.Alternatively, maybe I can think of the square as being divided into eight smaller triangles of equal area. If each of the four main triangles is s²/8, then two of them would be s²/4. Yeah, that makes sense.So, I think the correct answer is 1/4, which is option B.