💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. The sequence is {a_n}, and it's given that a₁ + a₇ + a₁₃ equals π. Then, I need to find the value of tan(a₂ + a₁₂). The options are A: √3, B: -√3, C: ±√3, D: -√3/3.Hmm, arithmetic sequences have a common difference, right? So, each term increases by a fixed amount. Let me recall the formula for the nth term of an arithmetic sequence. It's a_n = a₁ + (n - 1)d, where d is the common difference.So, let me write down the terms mentioned in the problem:a₁ is just a₁.a₇ would be a₁ + 6d.a₁₃ is a₁ + 12d.Adding these together: a₁ + (a₁ + 6d) + (a₁ + 12d) = π.Let me compute that: 3a₁ + 18d = π.Hmm, that simplifies to a₁ + 6d = π/3. Wait, that's interesting because a₇ is a₁ + 6d. So, a₇ = π/3.Now, the problem asks for tan(a₂ + a₁₂). Let me find expressions for a₂ and a₁₂.a₂ is a₁ + d.a₁₂ is a₁ + 11d.Adding these together: (a₁ + d) + (a₁ + 11d) = 2a₁ + 12d.Wait, 2a₁ + 12d. Let me see if I can relate this to something I already know.From earlier, I had 3a₁ + 18d = π, which simplifies to a₁ + 6d = π/3. If I multiply both sides by 2, I get 2a₁ + 12d = 2π/3.So, 2a₁ + 12d is equal to 2π/3. Therefore, a₂ + a₁₂ = 2π/3.Now, I need to find tan(2π/3). Let me recall the unit circle. 2π/3 is in the second quadrant, where tangent is negative. The reference angle for 2π/3 is π/3, and tan(π/3) is √3. Therefore, tan(2π/3) is -√3.So, the value of tan(a₂ + a₁₂) is -√3, which corresponds to option B.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I used the formula for the nth term correctly: a_n = a₁ + (n - 1)d.2. I correctly identified a₇ as a₁ + 6d.3. Adding a₁ + a₇ + a₁₃ gave me 3a₁ + 18d = π, which simplifies to a₁ + 6d = π/3, so a₇ = π/3.4. Then, for a₂ + a₁₂, I correctly found that it's 2a₁ + 12d, which is twice (a₁ + 6d), so 2*(π/3) = 2π/3.5. Calculating tan(2π/3) as -√3 is correct because tangent is negative in the second quadrant and the reference angle is π/3.Everything seems to check out. I think my answer is correct.**Final Answer**boxed{B}

💡Alright, so I've got this problem about an ellipse, and I need to find the range of its eccentricity. Let me try to unpack the problem step by step.First, the equation of the ellipse is given as (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a > b > 0). I remember that for an ellipse, the standard form is like this, with (a) being the semi-major axis and (b) the semi-minor axis. The foci of the ellipse are located at ((pm c, 0)), where (c) is the distance from the center to each focus. I also recall that (c = ae), where (e) is the eccentricity of the ellipse, and (e = sqrt{1 - frac{b^2}{a^2}}). So, the eccentricity is always between 0 and 1 for an ellipse.The problem mentions the right focus (F(c, 0)) and the upper vertex (A(0, b)). There's also a point (P) on the line (x = frac{a^2}{c}). So, the coordinates of (P) must be (left(frac{a^2}{c}, yright)) for some (y).The key condition given is that the dot product ((overrightarrow{FP} + overrightarrow{FA}) cdot overrightarrow{AP} = 0). I need to translate this vector condition into an equation involving coordinates and then solve for the eccentricity (e).Let me break this down:1. **Vectors (overrightarrow{FP}) and (overrightarrow{FA}):** - (overrightarrow{FP}) is the vector from (F) to (P). Since (F) is at ((c, 0)) and (P) is at (left(frac{a^2}{c}, yright)), the vector (overrightarrow{FP}) is (left(frac{a^2}{c} - c, y - 0right) = left(frac{a^2 - c^2}{c}, yright)). - (overrightarrow{FA}) is the vector from (F) to (A). (A) is at ((0, b)), so (overrightarrow{FA}) is ((0 - c, b - 0) = (-c, b)).2. **Sum of vectors (overrightarrow{FP} + overrightarrow{FA}):** - Adding the two vectors component-wise: - The x-component: (frac{a^2 - c^2}{c} + (-c) = frac{a^2 - c^2}{c} - c = frac{a^2 - c^2 - c^2}{c} = frac{a^2 - 2c^2}{c}). - The y-component: (y + b). - So, (overrightarrow{FP} + overrightarrow{FA} = left(frac{a^2 - 2c^2}{c}, y + bright)).3. **Vector (overrightarrow{AP}):** - (overrightarrow{AP}) is the vector from (A) to (P). (A) is at ((0, b)), so (overrightarrow{AP}) is (left(frac{a^2}{c} - 0, y - bright) = left(frac{a^2}{c}, y - bright)).4. **Dot Product Condition:** - The dot product of (overrightarrow{FP} + overrightarrow{FA}) and (overrightarrow{AP}) is zero: [ left(frac{a^2 - 2c^2}{c}right) cdot left(frac{a^2}{c}right) + (y + b) cdot (y - b) = 0 ] - Let me compute each part: - First term: (frac{(a^2 - 2c^2)a^2}{c^2}) - Second term: ((y + b)(y - b) = y^2 - b^2) - So, putting it together: [ frac{a^4 - 2a^2c^2}{c^2} + y^2 - b^2 = 0 ] - Simplify the first fraction: [ frac{a^4}{c^2} - 2a^2 + y^2 - b^2 = 0 ] - Rearranging: [ y^2 = -frac{a^4}{c^2} + 2a^2 + b^2 ] - Since (y^2) must be non-negative (as it's a square of a real number), the right-hand side must also be non-negative: [ -frac{a^4}{c^2} + 2a^2 + b^2 geq 0 ] - Let me rewrite this inequality: [ 2a^2 + b^2 geq frac{a^4}{c^2} ] - Multiply both sides by (c^2) (since (c^2 > 0)): [ (2a^2 + b^2)c^2 geq a^4 ] - Let's express everything in terms of (a) and (e), since (c = ae). So, (c^2 = a^2e^2). - Substitute (c^2) into the inequality: [ (2a^2 + b^2)a^2e^2 geq a^4 ] - Divide both sides by (a^2) (since (a > 0)): [ (2 + frac{b^2}{a^2})e^2 geq a^2 ] - Wait, that doesn't seem right. Let me check my substitution again.Wait, I think I made a mistake in substituting (c^2). Let's go back to the inequality:[(2a^2 + b^2)c^2 geq a^4]Since (c^2 = a^2 - b^2) (because for an ellipse, (c^2 = a^2 - b^2)), let me substitute that:[(2a^2 + b^2)(a^2 - b^2) geq a^4]Let me expand the left-hand side:[2a^2(a^2 - b^2) + b^2(a^2 - b^2) = 2a^4 - 2a^2b^2 + a^2b^2 - b^4 = 2a^4 - a^2b^2 - b^4]So, the inequality becomes:[2a^4 - a^2b^2 - b^4 geq a^4]Subtract (a^4) from both sides:[a^4 - a^2b^2 - b^4 geq 0]Hmm, this is a quartic equation. Maybe I can factor it or express it in terms of (e).Since (c^2 = a^2 - b^2), we can express (b^2 = a^2(1 - e^2)). Let me substitute (b^2) into the inequality:[a^4 - a^2(a^2(1 - e^2)) - (a^2(1 - e^2))^2 geq 0]Simplify each term:1. (a^4)2. (-a^2 cdot a^2(1 - e^2) = -a^4(1 - e^2))3. (-(a^2(1 - e^2))^2 = -a^4(1 - e^2)^2)So, putting it all together:[a^4 - a^4(1 - e^2) - a^4(1 - e^2)^2 geq 0]Factor out (a^4):[a^4left[1 - (1 - e^2) - (1 - e^2)^2right] geq 0]Since (a^4 > 0), we can divide both sides by (a^4):[1 - (1 - e^2) - (1 - e^2)^2 geq 0]Simplify the expression inside the brackets:First, (1 - (1 - e^2) = e^2).Then, subtract ((1 - e^2)^2):[e^2 - (1 - 2e^2 + e^4) = e^2 - 1 + 2e^2 - e^4 = 3e^2 - 1 - e^4]So, the inequality becomes:[3e^2 - 1 - e^4 geq 0]Let me rearrange it:[-e^4 + 3e^2 - 1 geq 0]Multiply both sides by (-1) (which reverses the inequality):[e^4 - 3e^2 + 1 leq 0]Now, this is a quartic inequality in terms of (e). Let me set (u = e^2), so the inequality becomes:[u^2 - 3u + 1 leq 0]This is a quadratic in (u). Let's find its roots:[u = frac{3 pm sqrt{9 - 4}}{2} = frac{3 pm sqrt{5}}{2}]So, the roots are (u = frac{3 + sqrt{5}}{2}) and (u = frac{3 - sqrt{5}}{2}).Since the coefficient of (u^2) is positive, the quadratic opens upwards. Therefore, the inequality (u^2 - 3u + 1 leq 0) holds between the roots:[frac{3 - sqrt{5}}{2} leq u leq frac{3 + sqrt{5}}{2}]But (u = e^2), and since (e) is the eccentricity of an ellipse, (0 < e < 1). Therefore, (0 < u < 1).So, we need to find the intersection of (frac{3 - sqrt{5}}{2} leq u leq frac{3 + sqrt{5}}{2}) with (0 < u < 1).First, compute the numerical values:- (frac{3 - sqrt{5}}{2} approx frac{3 - 2.236}{2} = frac{0.764}{2} = 0.382)- (frac{3 + sqrt{5}}{2} approx frac{3 + 2.236}{2} = frac{5.236}{2} = 2.618)So, the inequality (frac{3 - sqrt{5}}{2} leq u leq frac{3 + sqrt{5}}{2}) translates to approximately (0.382 leq u leq 2.618). But since (u = e^2 < 1), the upper bound is actually 1. Therefore, the valid range for (u) is:[frac{3 - sqrt{5}}{2} leq u < 1]Taking square roots to find (e):[sqrt{frac{3 - sqrt{5}}{2}} leq e < 1]Compute (sqrt{frac{3 - sqrt{5}}{2}}):Let me compute (frac{3 - sqrt{5}}{2}):[frac{3 - sqrt{5}}{2} approx frac{3 - 2.236}{2} = frac{0.764}{2} = 0.382]So, (sqrt{0.382} approx 0.618).Wait, 0.618 is approximately (frac{sqrt{5} - 1}{2}), since (sqrt{5} approx 2.236), so (sqrt{5} - 1 approx 1.236), and divided by 2 is approximately 0.618.Yes, indeed:[sqrt{frac{3 - sqrt{5}}{2}} = frac{sqrt{5} - 1}{2}]Let me verify that:Let me square (frac{sqrt{5} - 1}{2}):[left(frac{sqrt{5} - 1}{2}right)^2 = frac{5 - 2sqrt{5} + 1}{4} = frac{6 - 2sqrt{5}}{4} = frac{3 - sqrt{5}}{2}]Yes, that's correct. So,[sqrt{frac{3 - sqrt{5}}{2}} = frac{sqrt{5} - 1}{2}]Therefore, the range for (e) is:[frac{sqrt{5} - 1}{2} leq e < 1]So, the eccentricity (e) must be at least (frac{sqrt{5} - 1}{2}) and less than 1.Now, let me check the options provided:A: ([1/2, 1))B: ([sqrt{2}/2, 1))C: ([frac{sqrt{5} - 1}{2}, 1))D: ((0, sqrt{2}/2])So, option C matches our result.But just to be thorough, let me compute (frac{sqrt{5} - 1}{2}):[sqrt{5} approx 2.236, so sqrt{5} - 1 approx 1.236, divided by 2 is approximately 0.618.]So, 0.618 is approximately the lower bound. Comparing with the options:- A starts at 0.5- B starts at approximately 0.707- C starts at approximately 0.618- D ends at approximately 0.707So, our result is C.I think that's solid. I don't see any mistakes in the calculations now.**Final Answer**boxed{C}

💡好，我现在要解决这个几何问题。题目是说，一个锐角三角形ABC，满足AB > AC，O是外心，H是垂心。BH交AC于B₁，CH交AB于C₁。已知OH平行于B₁C₁，要证明cos2B + cos2C + 1 = 0。首先，我需要画一个图来帮助理解问题。画出三角形ABC，AB比AC长，所以顶点B可能在左边，顶点C在右边，顶点A在上面。O是外心，H是垂心。BH和CH分别交AC和AB于B₁和C₁。接下来，已知OH平行于B₁C₁，这是一个重要的条件，可能需要用到平行线的性质，比如对应角相等或者相似三角形的性质。首先，我应该回顾一下外心和垂心的性质。外心O是三角形ABC的外接圆的圆心，是三条边的垂直平分线的交点。垂心H是三条高线的交点。接下来，考虑点B₁和C₁的位置。BH交AC于B₁，CH交AB于C₁。这意味着B₁在AC上，C₁在AB上。接下来，OH平行于B₁C₁，这可能意味着这两条线段之间存在某种比例关系或者角度关系。接下来，我应该考虑连接一些辅助线，比如连接AO和AH，因为O和H都是与A相关的点。连接AO和AH可能会形成一些有用的三角形或者角度关系。然后，考虑到A、B₁、H、C₁四点共圆，可能是因为它们在某个圆上，这可能是因为它们满足某种圆的性质，比如对角和为180度。或者，可能是因为某些角度相等，导致它们共圆。接下来，我需要分析角度关系。因为OH平行于B₁C₁，所以它们的对应角相等。比如，∠OHC₁可能等于∠HC₁B₁，或者类似的角相等。接下来，我需要考虑四点共圆的情况。比如，B、C、B₁、C₁四点是否共圆？如果是的话，那么它们的对角和为180度，或者某些角相等，这可能帮助我找到角度之间的关系。然后，我需要考虑三角形的外接圆半径R，以及AH的长度。因为AH是从A到H的长度，可以用外接圆半径和角度来表示，比如AH = 2R cos A。接下来，我需要分析AO和AH之间的关系，以及它们与OH之间的角度关系。可能涉及到余弦定理或者正弦定理来表达这些长度之间的关系。然后，我可能需要使用一些三角恒等式，比如cos(A + B)或者cos(A - B)，来将这些角度关系转化为代数表达式，最终得到cos2B + cos2C + 1 = 0。在这个过程中，我可能会遇到一些困难，比如如何证明四点共圆，或者如何正确应用角度关系。我需要仔细检查每一步，确保逻辑正确，没有遗漏任何条件。最后，我需要将所有这些分析整合起来，形成一个完整的证明过程，从已知条件出发，逐步推导出所需的结论。总结一下，我的思路是：1. 画图并标出所有已知点和线。2. 连接辅助线，如AO和AH。3. 利用OH平行于B₁C₁的条件，分析角度关系。4. 利用四点共圆的性质，寻找角度之间的关系。5. 应用三角恒等式和外接圆半径的关系，推导出所需的结论。现在，我需要一步步地展开这些思路，确保每一步都正确无误。

💡Alright, so I have these two fractions to simplify. Let me take them one at a time and see if I can figure out how to approach each one.Starting with the first one: (frac{a sqrt{x} - x sqrt{a}}{sqrt{a} - sqrt{x}}). Hmm, okay. Both the numerator and the denominator have square roots, so maybe I can factor something out or find a common term. Let me look at the numerator first: (a sqrt{x} - x sqrt{a}). I notice that both terms have a square root, but one has (a) and the other has (x). Maybe I can factor out a common term. Let's see, if I factor out (sqrt{a}) from the first term, I get (sqrt{a} cdot sqrt{x}), which is (sqrt{a x}). Similarly, if I factor out (sqrt{x}) from the second term, I get (sqrt{x} cdot sqrt{a}), which is also (sqrt{a x}). So, both terms in the numerator have a factor of (sqrt{a x}). Let me write that out:(a sqrt{x} - x sqrt{a} = sqrt{a x} cdot sqrt{a} - sqrt{a x} cdot sqrt{x})Wait, that doesn't seem right. Let me try again. If I factor out (sqrt{a x}) from both terms, it should look like:(a sqrt{x} - x sqrt{a} = sqrt{a x} cdot (sqrt{a} - sqrt{x}))Yes, that makes sense because:(sqrt{a x} cdot sqrt{a} = a sqrt{x})and(sqrt{a x} cdot sqrt{x} = x sqrt{a})So, the numerator becomes (sqrt{a x} (sqrt{a} - sqrt{x})). Now, looking back at the original fraction:(frac{a sqrt{x} - x sqrt{a}}{sqrt{a} - sqrt{x}} = frac{sqrt{a x} (sqrt{a} - sqrt{x})}{sqrt{a} - sqrt{x}})Oh, I see! The ((sqrt{a} - sqrt{x})) terms in the numerator and denominator cancel each other out, leaving just (sqrt{a x}). So, the simplified form of the first fraction is (sqrt{a x}).Alright, that wasn't too bad. Now, moving on to the second fraction: (frac{sqrt{a + b} - sqrt{a - b}}{a + b - sqrt{a^2 - b^2}}). This one looks a bit trickier. Let me see if I can find a way to simplify the denominator first.The denominator is (a + b - sqrt{a^2 - b^2}). I remember that (a^2 - b^2) can be factored into ((a + b)(a - b)). So, (sqrt{a^2 - b^2} = sqrt{(a + b)(a - b)}). That might be useful.Let me rewrite the denominator using this:(a + b - sqrt{(a + b)(a - b)})Hmm, I notice that both terms in the denominator have a factor of (sqrt{a + b}). Let me factor that out:(sqrt{a + b} cdot sqrt{a + b} - sqrt{a + b} cdot sqrt{a - b})Which simplifies to:(sqrt{a + b} (sqrt{a + b} - sqrt{a - b}))So, the denominator becomes (sqrt{a + b} (sqrt{a + b} - sqrt{a - b})). Now, looking back at the original fraction:(frac{sqrt{a + b} - sqrt{a - b}}{a + b - sqrt{a^2 - b^2}} = frac{sqrt{a + b} - sqrt{a - b}}{sqrt{a + b} (sqrt{a + b} - sqrt{a - b})})I see that the ((sqrt{a + b} - sqrt{a - b})) terms in the numerator and denominator cancel each other out, leaving just (frac{1}{sqrt{a + b}}). So, the simplified form of the second fraction is (frac{1}{sqrt{a + b}}).Wait a second, let me double-check that. If I have (frac{sqrt{a + b} - sqrt{a - b}}{sqrt{a + b} (sqrt{a + b} - sqrt{a - b})}), then yes, the ((sqrt{a + b} - sqrt{a - b})) terms do cancel, and I'm left with (frac{1}{sqrt{a + b}}). That seems correct.Okay, so both fractions simplified nicely by factoring out common terms and then canceling them out. I think I've got it!

💡First, I need to determine the number of intervals between the trees. Since there are eight trees, there are 7 intervals from the first to the last tree.Next, I know that the distance from the first tree to the fifth tree is 100 feet. There are 4 intervals between the first and fifth trees. By dividing the total distance by the number of intervals, I can find the distance between each consecutive tree:[text{Distance between consecutive trees} = frac{100 text{ feet}}{4} = 25 text{ feet}]Finally, to find the total distance from the first to the last tree, I multiply the distance between each pair of consecutive trees by the total number of intervals:[text{Total distance} = 25 text{ feet/space} times 7 text{ spaces} = 175 text{ feet}]The distance between the first and last trees is 175 feet.

💡Okay, so I have these two equations: 2x minus 3y equals 24, and x plus 2y equals 15. I need to find the value of y. Hmm, let me think about how to approach this. I remember from class that when you have two equations with two variables, you can solve them using either substitution or elimination. I think elimination might be easier here because the coefficients look manageable.First, I'll write down the equations again to make sure I have them right:1. 2x - 3y = 242. x + 2y = 15Alright, so I need to eliminate one of the variables. Let's see, if I can make the coefficients of x the same in both equations, I can subtract one equation from the other to eliminate x. To do that, I'll multiply the second equation by 2. That way, the coefficient of x in both equations will be 2.So, multiplying the second equation by 2:2*(x + 2y) = 2*15Which simplifies to:2x + 4y = 30Now I have the two equations:1. 2x - 3y = 242. 2x + 4y = 30Next, I'll subtract the first equation from the second equation to eliminate x. Let's set that up:(2x + 4y) - (2x - 3y) = 30 - 24Simplifying the left side:2x + 4y - 2x + 3y = 6The 2x and -2x cancel out, leaving:7y = 6Now, to solve for y, I'll divide both sides by 7:y = 6/7Wait, that seems a bit odd. Let me double-check my steps to make sure I didn't make a mistake. Starting from the beginning:Original equations:2x - 3y = 24x + 2y = 15I multiplied the second equation by 2 to get:2x + 4y = 30Subtracting the first equation from this:(2x + 4y) - (2x - 3y) = 30 - 242x + 4y - 2x + 3y = 67y = 6y = 6/7Hmm, that seems consistent. Maybe I was just surprised because I expected a whole number, but fractions are okay too. Let me check by plugging y back into one of the original equations to find x and see if it makes sense.Let's use the second original equation:x + 2y = 15Substituting y = 6/7:x + 2*(6/7) = 15x + 12/7 = 15To solve for x, subtract 12/7 from both sides:x = 15 - 12/7Convert 15 to sevenths:15 = 105/7So, x = 105/7 - 12/7 = 93/7Now, let's check if these values satisfy the first equation:2x - 3y = 24Substituting x = 93/7 and y = 6/7:2*(93/7) - 3*(6/7) = 24186/7 - 18/7 = 24(186 - 18)/7 = 24168/7 = 2424 = 24Okay, that checks out. So even though y is a fraction, it's correct. I guess I was just expecting a whole number, but math doesn't care about our expectations! So, y is indeed 6/7.