💡Okay, so I have this problem about an arithmetic progression. It says that the sum of the fourth and twelfth terms is 20, and I need to find the sum of the first 10 terms. Hmm, arithmetic progressions... I remember they have a common difference between consecutive terms. Let me try to recall the formulas.First, the nth term of an arithmetic progression (AP) is given by ( a_n = a + (n - 1)d ), where ( a ) is the first term and ( d ) is the common difference. So, the fourth term would be ( a_4 = a + 3d ) and the twelfth term would be ( a_{12} = a + 11d ). The problem states that the sum of these two terms is 20. So, if I add them together: ( a_4 + a_{12} = (a + 3d) + (a + 11d) = 2a + 14d = 20 ). Simplifying that, I get ( a + 7d = 10 ). Okay, so that's one equation involving ( a ) and ( d ).Now, I need to find the sum of the first 10 terms. The formula for the sum of the first ( n ) terms of an AP is ( S_n = frac{n}{2} [2a + (n - 1)d] ). Plugging in ( n = 10 ), I get ( S_{10} = frac{10}{2} [2a + 9d] = 5[2a + 9d] ).But wait, from the earlier equation, I have ( a + 7d = 10 ). Maybe I can express ( a ) in terms of ( d ) or vice versa. Let's solve for ( a ): ( a = 10 - 7d ). Now, substitute this into the sum formula: ( S_{10} = 5[2(10 - 7d) + 9d] = 5[20 - 14d + 9d] = 5[20 - 5d] = 100 - 25d ). Hmm, so the sum depends on ( d ). But I don't know the value of ( d ). Is there a way to find ( d ) from the given information? Let me think. I only have one equation ( a + 7d = 10 ), and I need another equation to solve for both ( a ) and ( d ). But the problem doesn't provide any additional information. Wait, maybe I don't need the exact values of ( a ) and ( d ). Let me look back at the sum formula: ( S_{10} = 100 - 25d ). If I can express this in terms of the known sum ( a + 7d = 10 ), maybe I can find a relationship. Let me see, ( a + 7d = 10 ) implies that ( 2a + 14d = 20 ). Comparing this with the sum formula, which is ( S_{10} = 100 - 25d ), I notice that ( 2a + 14d = 20 ) is part of the sum formula. Wait, in the sum formula, I have ( 2a + 9d ). If I can express ( 2a + 9d ) in terms of ( 2a + 14d ), maybe I can find ( S_{10} ). Let's try:From ( 2a + 14d = 20 ), subtract ( 5d ) from both sides: ( 2a + 9d = 20 - 5d ). So, ( S_{10} = 5[20 - 5d] = 100 - 25d ). But I still don't know ( d ). Is there a way to find ( d ) without additional information? Maybe not directly. But perhaps I can express ( S_{10} ) in terms of the known sum ( a + 7d = 10 ).Let me think differently. If ( a + 7d = 10 ), then the seventh term ( a_7 = a + 6d ) is close to this. Wait, no, ( a_7 = a + 6d ), which is one less than ( a + 7d ). So, ( a_7 = 10 - d ).But I'm not sure if that helps. Maybe I can think about the average of the first 10 terms. The sum ( S_{10} ) is also equal to the average of the first and tenth term multiplied by 10. The tenth term is ( a_{10} = a + 9d ). So, the average is ( frac{a + (a + 9d)}{2} = frac{2a + 9d}{2} ). Therefore, ( S_{10} = 10 times frac{2a + 9d}{2} = 5(2a + 9d) ), which is the same as before.I still end up with ( S_{10} = 100 - 25d ). Without knowing ( d ), I can't find the exact value of ( S_{10} ). Maybe there's a different approach. Let me consider that in an AP, the sum of terms equidistant from the ends is constant. For example, ( a_1 + a_{10} = a_2 + a_9 = ... = a_5 + a_6 ). But I'm not sure how that helps here since I don't have information about other terms. Alternatively, maybe I can express ( S_{10} ) in terms of ( a + 7d ). Since ( a + 7d = 10 ), perhaps I can relate ( 2a + 9d ) to this.Let me write ( 2a + 9d = 2(a + 7d) - 5d = 2(10) - 5d = 20 - 5d ). So, ( S_{10} = 5(20 - 5d) = 100 - 25d ). But again, without knowing ( d ), I can't find the exact sum. Maybe the problem expects an expression in terms of ( d ), but the question asks for the sum, implying a numerical answer. Perhaps I missed something.Wait, maybe I can find ( d ) by considering that the sum of the fourth and twelfth terms is 20. Let me write that again: ( a_4 + a_{12} = 20 ). Which is ( (a + 3d) + (a + 11d) = 2a + 14d = 20 ). So, ( a + 7d = 10 ). If I think about the sum of the first 10 terms, ( S_{10} = frac{10}{2} [2a + 9d] = 5(2a + 9d) ). I can express ( 2a + 9d ) as ( 2(a + 7d) - 5d = 2(10) - 5d = 20 - 5d ). So, ( S_{10} = 5(20 - 5d) = 100 - 25d ).But I still don't know ( d ). Is there a way to find ( d ) without additional information? It seems like I need another equation. Maybe I can assume something about the AP, but that's not good practice. Alternatively, perhaps the problem expects me to recognize that the sum of the first 10 terms is 10 times the average of the fourth and twelfth terms.Wait, the average of the fourth and twelfth terms is ( frac{a_4 + a_{12}}{2} = frac{20}{2} = 10 ). In an AP, the average of two terms equidistant from the ends is equal to the average of the entire sequence. But the fourth and twelfth terms are not equidistant from the ends of the first 10 terms. The fourth term is the fourth, and the twelfth term is beyond the tenth term. So, that might not help.Alternatively, maybe I can think about the sum of the first 10 terms in terms of the fourth and twelfth terms. Let me see:The sum of the first 10 terms is ( S_{10} = a_1 + a_2 + ... + a_{10} ). I know ( a_4 + a_{12} = 20 ). But ( a_{12} ) is beyond the tenth term, so it's not directly part of ( S_{10} ). Wait, maybe I can express ( a_{12} ) in terms of ( a_{10} ). Since ( a_{12} = a_{10} + 2d ). So, ( a_4 + a_{12} = a_4 + a_{10} + 2d = 20 ). But ( a_4 = a + 3d ) and ( a_{10} = a + 9d ). So, ( (a + 3d) + (a + 9d) + 2d = 2a + 14d + 2d = 2a + 16d = 20 ). Wait, that's different from before. Did I make a mistake?No, earlier I had ( a_4 + a_{12} = 2a + 14d = 20 ). Now, expressing ( a_{12} ) as ( a_{10} + 2d ), I get ( a_4 + a_{10} + 2d = 20 ). But ( a_4 + a_{10} = (a + 3d) + (a + 9d) = 2a + 12d ). So, ( 2a + 12d + 2d = 2a + 14d = 20 ), which matches the earlier equation. So, no new information.I'm stuck because I have one equation and two unknowns. Maybe the problem expects me to realize that the sum of the first 10 terms can be expressed in terms of the given sum, but I'm not sure. Alternatively, perhaps there's a property of APs that I'm missing.Wait, let's think about the symmetry in the terms. The fourth term is ( a + 3d ) and the twelfth term is ( a + 11d ). The average of these two terms is ( frac{(a + 3d) + (a + 11d)}{2} = frac{2a + 14d}{2} = a + 7d = 10 ). So, the average of the fourth and twelfth terms is 10.In an AP, the average of two terms equidistant from the ends is equal to the average of the entire sequence. But in this case, the fourth and twelfth terms are not equidistant from the ends of the first 10 terms. The fourth term is the fourth, and the twelfth term is beyond the tenth term. So, maybe that's not helpful.Alternatively, perhaps I can think about the sum of the first 10 terms in terms of the average of the fourth and twelfth terms. Since the average is 10, and there are 10 terms, maybe the sum is 10 times 10, which is 100. But that seems too simplistic and ignores the common difference.Wait, let me check. If the average of the fourth and twelfth terms is 10, and if the AP is symmetric around the seventh term, then maybe the average of the entire first 10 terms is also 10. But I'm not sure if that's the case.Alternatively, maybe I can express the sum of the first 10 terms in terms of the average of the fourth and twelfth terms. Since ( a + 7d = 10 ), and the sum of the first 10 terms is ( 5(2a + 9d) ), which is ( 5(2(a + 7d) - 5d) = 5(20 - 5d) = 100 - 25d ). But without knowing ( d ), I can't find the exact sum. Maybe the problem expects me to recognize that the sum is 100, assuming ( d = 0 ), but that would make it a constant sequence, which is a special case of an AP. But I don't think that's the intended approach.Alternatively, maybe I can express the sum in terms of the given sum. Since ( a + 7d = 10 ), and the sum of the first 10 terms is ( 100 - 25d ), perhaps I can express ( d ) in terms of ( a ) and substitute back. But that would just give me ( S_{10} = 100 - 25d ), which doesn't help.Wait, maybe I can find ( d ) by considering that the sum of the fourth and twelfth terms is 20, and the sum of the first 10 terms is related to this. Let me think about the relationship between these sums.The sum of the fourth and twelfth terms is 20, which is ( 2a + 14d = 20 ). The sum of the first 10 terms is ( 10a + 45d ). If I can express ( 10a + 45d ) in terms of ( 2a + 14d ), maybe I can find a relationship.Let me see, ( 10a + 45d = 5(2a + 9d) ). From ( 2a + 14d = 20 ), I can write ( 2a = 20 - 14d ). Substituting into ( 2a + 9d ), I get ( (20 - 14d) + 9d = 20 - 5d ). So, ( S_{10} = 5(20 - 5d) = 100 - 25d ).But again, I'm stuck because I don't know ( d ). Maybe there's a way to find ( d ) using the fact that the sum of the first 10 terms is related to the sum of the fourth and twelfth terms. Let me think about the relationship between these sums.If I consider the sum of the first 10 terms, ( S_{10} = 10a + 45d ), and I know ( 2a + 14d = 20 ), I can solve for ( a ) in terms of ( d ): ( a = 10 - 7d ). Substituting into ( S_{10} ), I get ( S_{10} = 10(10 - 7d) + 45d = 100 - 70d + 45d = 100 - 25d ).So, ( S_{10} = 100 - 25d ). But without knowing ( d ), I can't find the exact value. Maybe the problem expects me to realize that the sum is 100, but that would only be true if ( d = 0 ), which isn't necessarily the case.Alternatively, perhaps I can express ( S_{10} ) in terms of the given sum. Since ( 2a + 14d = 20 ), and ( S_{10} = 10a + 45d ), I can write ( S_{10} = 5(2a + 9d) ). From ( 2a + 14d = 20 ), I can express ( 2a = 20 - 14d ), so ( 2a + 9d = 20 - 14d + 9d = 20 - 5d ). Therefore, ( S_{10} = 5(20 - 5d) = 100 - 25d ).But I still don't know ( d ). Maybe I can find ( d ) by considering that the sum of the first 10 terms must be a multiple of 5, given the formula ( 100 - 25d ). But that doesn't help me find ( d ).Wait, maybe I can think about the problem differently. If I consider that the sum of the fourth and twelfth terms is 20, and I need the sum of the first 10 terms, perhaps there's a relationship between these sums that doesn't require knowing ( a ) and ( d ).Let me consider the sum of the first 10 terms: ( S_{10} = frac{10}{2} [2a + 9d] = 5(2a + 9d) ). I know that ( 2a + 14d = 20 ), so ( 2a = 20 - 14d ). Substituting into ( 2a + 9d ), I get ( 20 - 14d + 9d = 20 - 5d ). Therefore, ( S_{10} = 5(20 - 5d) = 100 - 25d ).But I still don't know ( d ). Maybe I can express ( d ) in terms of ( S_{10} ). Let me rearrange the equation: ( S_{10} = 100 - 25d ) implies ( d = frac{100 - S_{10}}{25} ). But that doesn't help me find ( S_{10} ) without knowing ( d ).I'm stuck because I have one equation and two unknowns. Maybe the problem expects me to recognize that the sum of the first 10 terms is 100, but that's only true if ( d = 0 ), which isn't necessarily the case. Alternatively, maybe I made a mistake in my earlier steps.Let me double-check my calculations. Starting from ( a_4 + a_{12} = 20 ):( a_4 = a + 3d )( a_{12} = a + 11d )Sum: ( (a + 3d) + (a + 11d) = 2a + 14d = 20 )Simplify: ( a + 7d = 10 )Sum of first 10 terms: ( S_{10} = frac{10}{2} [2a + 9d] = 5(2a + 9d) )Express ( 2a + 9d ) in terms of ( a + 7d ):( 2a + 9d = 2(a + 7d) - 5d = 2(10) - 5d = 20 - 5d )Therefore, ( S_{10} = 5(20 - 5d) = 100 - 25d )Yes, that seems correct. So, without knowing ( d ), I can't find the exact value of ( S_{10} ). Maybe the problem expects me to express the sum in terms of ( d ), but the question asks for the sum, implying a numerical answer. Perhaps I missed something in the problem statement.Wait, the problem says "the sum of the fourth and twelfth term of an arithmetic progression is 20." It doesn't provide any other information. So, with the given information, it's impossible to determine the exact sum of the first 10 terms because there are infinitely many APs that satisfy ( a + 7d = 10 ) with different values of ( d ), leading to different sums for the first 10 terms.Therefore, the problem might be incomplete or there might be a standard assumption I'm missing. Alternatively, maybe the problem expects me to recognize that the sum of the first 10 terms is 100, assuming ( d = 0 ), but that's not necessarily correct.Wait, if ( d = 0 ), then all terms are equal to ( a ), and ( a + 7(0) = 10 ) implies ( a = 10 ). Then, the sum of the first 10 terms would be ( 10 times 10 = 100 ). But if ( d ) is not zero, the sum would be different. For example, if ( d = 1 ), then ( a = 10 - 7(1) = 3 ), and ( S_{10} = 100 - 25(1) = 75 ). If ( d = 2 ), ( a = 10 - 14 = -4 ), and ( S_{10} = 100 - 50 = 50 ). So, the sum varies depending on ( d ).Therefore, without additional information, the sum of the first 10 terms cannot be determined uniquely. The problem might need more information, such as another term or the common difference, to find a unique solution.Alternatively, maybe the problem expects me to realize that the sum of the first 10 terms is 100, but that's only true if ( d = 0 ), which isn't necessarily the case. So, I think the problem is incomplete or there's a missing piece of information.Wait, maybe I can express the sum in terms of the given sum. Since ( a + 7d = 10 ), and ( S_{10} = 100 - 25d ), I can write ( S_{10} = 100 - 25d ). But without knowing ( d ), I can't find the exact value. Maybe the problem expects me to express the sum in terms of ( d ), but the question asks for the sum, implying a numerical answer.Alternatively, perhaps I can find ( d ) by considering that the sum of the first 10 terms must be a multiple of 5, given the formula ( 100 - 25d ). But that doesn't help me find ( d ).I'm stuck because I have one equation and two unknowns. Maybe the problem expects me to recognize that the sum of the first 10 terms is 100, but that's only true if ( d = 0 ), which isn't necessarily the case. Alternatively, maybe I made a mistake in my earlier steps.Wait, let me try a different approach. Let's assume that the sum of the first 10 terms is ( S ). Then, ( S = 10a + 45d ). From the given information, ( 2a + 14d = 20 ), so ( a = 10 - 7d ). Substituting into ( S ), I get ( S = 10(10 - 7d) + 45d = 100 - 70d + 45d = 100 - 25d ).So, ( S = 100 - 25d ). But I still don't know ( d ). Maybe I can express ( d ) in terms of ( S ): ( d = frac{100 - S}{25} ). But that doesn't help me find ( S ) without knowing ( d ).I think I've exhausted all possible approaches with the given information. Without additional information about the AP, such as another term or the common difference, I can't determine the exact sum of the first 10 terms. The problem might need more details to have a unique solution.Alternatively, maybe the problem expects me to realize that the sum of the first 10 terms is 100, assuming ( d = 0 ), but that's not necessarily correct. So, I think the answer is that the sum of the first 10 terms cannot be determined with the given information.Wait, but the problem asks for the sum, so maybe I need to express it in terms of the given sum. Since ( a + 7d = 10 ), and ( S_{10} = 100 - 25d ), I can write ( S_{10} = 100 - 25d ). But without knowing ( d ), I can't find the exact value. Maybe the problem expects me to express the sum in terms of ( d ), but the question asks for the sum, implying a numerical answer.I think I've tried all possible methods, and without additional information, the sum of the first 10 terms cannot be uniquely determined. Therefore, the problem might be incomplete or there might be a standard assumption I'm missing.

💡First, I need to evaluate the expression (2 + 3 cdot 4 - 5 + 6 div 2).I'll start by performing the multiplication and division from left to right. Calculating (3 cdot 4) gives me 12, and (6 div 2) equals 3.Substituting these values back into the expression, it becomes (2 + 12 - 5 + 3).Next, I'll perform the addition and subtraction from left to right.Adding 2 and 12 gives me 14. Subtracting 5 from 14 results in 9. Finally, adding 3 to 9 gives me the final answer of 12.

💡First, I need to determine the total distance the train travels while crossing the bridge. This distance is the sum of the train's length and the bridge's length.The train's speed is given in kilometers per hour (km/h), but the time is provided in seconds. To make the units consistent, I should convert the speed from km/h to meters per second (m/s). I'll use the conversion factor: 1 km/h = 1000 meters / 3600 seconds.Next, I'll calculate the total distance by multiplying the speed in m/s by the time in seconds.Finally, I'll subtract the length of the train from the total distance to find the length of the bridge.

💡Okay, so I have this problem where I'm given two initial values, a₀ = -2 and b₀ = 1. Then, for each n ≥ 0, there are recursive formulas to find a_{n+1} and b_{n+1}. The formulas are:a_{n+1} = a_n + b_n + √(a_n² + b_n²)b_{n+1} = a_n + b_n - √(a_n² + b_n²}And I need to find the value of 1/a_{2012} + 1/b_{2012}. Hmm, 2012 is a pretty large number, so calculating each term step by step up to that point isn't practical. There must be a pattern or some invariant that stays the same regardless of n. Maybe if I can find a relationship between a_n and b_n that remains constant, I can use that to find the desired value without computing all the terms.Let me start by computing the first few terms to see if I can spot a pattern.Starting with n = 0:a₀ = -2b₀ = 1Compute a₁ and b₁:a₁ = a₀ + b₀ + √(a₀² + b₀²) = (-2) + 1 + √((-2)² + 1²) = (-1) + √(4 + 1) = (-1) + √5Similarly, b₁ = a₀ + b₀ - √(a₀² + b₀²) = (-2) + 1 - √5 = (-1) - √5So, a₁ = -1 + √5 and b₁ = -1 - √5Now, let's compute 1/a₁ + 1/b₁:1/a₁ + 1/b₁ = 1/(-1 + √5) + 1/(-1 - √5)To simplify this, I can rationalize the denominators:1/(-1 + √5) = (-1 - √5)/[(-1 + √5)(-1 - √5)] = (-1 - √5)/(1 - 5) = (-1 - √5)/(-4) = (1 + √5)/4Similarly, 1/(-1 - √5) = (-1 + √5)/[(-1 - √5)(-1 + √5)] = (-1 + √5)/(1 - 5) = (-1 + √5)/(-4) = (1 - √5)/4Adding these together:(1 + √5)/4 + (1 - √5)/4 = [1 + √5 + 1 - √5]/4 = 2/4 = 1/2Interesting! So, 1/a₁ + 1/b₁ = 1/2, which is the same as 1/a₀ + 1/b₀:1/a₀ + 1/b₀ = 1/(-2) + 1/1 = -1/2 + 1 = 1/2So, it seems like 1/a_n + 1/b_n is constant for n=0 and n=1. Maybe this is an invariant?Let me test it for n=2. Compute a₂ and b₂:a₂ = a₁ + b₁ + √(a₁² + b₁²)But wait, a₁ = -1 + √5 and b₁ = -1 - √5So, a₁ + b₁ = (-1 + √5) + (-1 - √5) = -2Now, compute √(a₁² + b₁²):a₁² = (-1 + √5)² = 1 - 2√5 + 5 = 6 - 2√5b₁² = (-1 - √5)² = 1 + 2√5 + 5 = 6 + 2√5So, a₁² + b₁² = (6 - 2√5) + (6 + 2√5) = 12Therefore, √(a₁² + b₁²) = √12 = 2√3So, a₂ = a₁ + b₁ + √(a₁² + b₁²) = (-2) + 2√3 = -2 + 2√3Similarly, b₂ = a₁ + b₁ - √(a₁² + b₁²) = (-2) - 2√3 = -2 - 2√3Now, compute 1/a₂ + 1/b₂:1/(-2 + 2√3) + 1/(-2 - 2√3)Again, let's rationalize:1/(-2 + 2√3) = (-2 - 2√3)/[(-2 + 2√3)(-2 - 2√3)] = (-2 - 2√3)/(4 - (2√3)²) = (-2 - 2√3)/(4 - 12) = (-2 - 2√3)/(-8) = (2 + 2√3)/8 = (1 + √3)/4Similarly, 1/(-2 - 2√3) = (-2 + 2√3)/[(-2 - 2√3)(-2 + 2√3)] = (-2 + 2√3)/(4 - (2√3)²) = (-2 + 2√3)/(4 - 12) = (-2 + 2√3)/(-8) = (2 - 2√3)/8 = (1 - √3)/4Adding these together:(1 + √3)/4 + (1 - √3)/4 = [1 + √3 + 1 - √3]/4 = 2/4 = 1/2Wow, again, 1/a₂ + 1/b₂ = 1/2. So, it seems like this is indeed an invariant. It stays the same for each n.Therefore, regardless of how many times we apply the recursion, the sum 1/a_n + 1/b_n remains 1/2. So, even for n=2012, this sum should still be 1/2.But just to make sure, let me try to generalize this. Suppose for some n, 1/a_n + 1/b_n = 1/2. Then, does this imply that 1/a_{n+1} + 1/b_{n+1} = 1/2 as well?Let me compute 1/a_{n+1} + 1/b_{n+1} in terms of a_n and b_n.Given:a_{n+1} = a_n + b_n + √(a_n² + b_n²)b_{n+1} = a_n + b_n - √(a_n² + b_n²)So, let me denote S_n = a_n + b_n and D_n = √(a_n² + b_n²)Therefore, a_{n+1} = S_n + D_n and b_{n+1} = S_n - D_nSo, 1/a_{n+1} + 1/b_{n+1} = 1/(S_n + D_n) + 1/(S_n - D_n)Let me compute this:= [ (S_n - D_n) + (S_n + D_n) ] / [ (S_n + D_n)(S_n - D_n) ]= [ 2S_n ] / [ S_n² - D_n² ]But D_n² = a_n² + b_n², so S_n² - D_n² = (a_n + b_n)² - (a_n² + b_n²) = 2a_n b_nTherefore, 1/a_{n+1} + 1/b_{n+1} = (2S_n) / (2a_n b_n) = S_n / (a_n b_n)But S_n = a_n + b_n, so:= (a_n + b_n) / (a_n b_n) = 1/a_n + 1/b_nSo, 1/a_{n+1} + 1/b_{n+1} = 1/a_n + 1/b_nThis shows that the sum 1/a_n + 1/b_n is indeed constant for all n. Therefore, it doesn't matter what n is; the value remains the same as the initial value.Given that, since 1/a₀ + 1/b₀ = 1/(-2) + 1/1 = -1/2 + 1 = 1/2, this means that for any n, including n=2012, the sum 1/a_n + 1/b_n is 1/2.Therefore, the answer is 1/2.**Final Answer**boxed{dfrac{1}{2}}

💡Okay, so I have this problem here: the line 2x + y - 2 = 0 passes through the upper vertex and the right focus of the ellipse x²/a² + y²/b² = 1, where a and b are positive. I need to find the equation of the ellipse. Hmm, let me think about how to approach this.First, I remember that for an ellipse in the form x²/a² + y²/b² = 1, the major and minor axes depend on whether a is greater than b or vice versa. Since the problem mentions the upper vertex and the right focus, I think this ellipse is oriented vertically because the upper vertex is along the y-axis. Wait, no, actually, the standard form is x²/a² + y²/b² = 1, so if a > b, it's wider along the x-axis, making the major axis horizontal. But if b > a, then the major axis is vertical. Hmm, the problem says a > 0 and b > 0, but doesn't specify which is larger. Maybe I need to figure that out.But the line passes through the upper vertex and the right focus. The upper vertex would be (0, b) if the major axis is vertical, or (0, a) if it's horizontal. Wait, no, for an ellipse, the vertices are at (±a, 0) for a horizontal major axis and (0, ±b) for a vertical major axis. Similarly, the foci are at (±c, 0) for horizontal and (0, ±c) for vertical, where c = sqrt(a² - b²) for horizontal and c = sqrt(b² - a²) for vertical.Wait, hold on, I might be mixing things up. Let me clarify:For an ellipse x²/a² + y²/b² = 1:- If a > b, it's a horizontal ellipse, with major axis along the x-axis, vertices at (±a, 0), and foci at (±c, 0), where c = sqrt(a² - b²).- If b > a, it's a vertical ellipse, with major axis along the y-axis, vertices at (0, ±b), and foci at (0, ±c), where c = sqrt(b² - a²).In this problem, the line passes through the upper vertex and the right focus. So, the upper vertex would be (0, b) if it's a vertical ellipse, and the right focus would be (c, 0) if it's a horizontal ellipse. Wait, but if it's a vertical ellipse, the foci are along the y-axis, so the right focus wouldn't make sense. Similarly, if it's a horizontal ellipse, the upper vertex would be (0, b), but if a > b, then b < a, so (0, b) is just a co-vertex, not the upper vertex. Hmm, maybe I need to think differently.Wait, perhaps the ellipse is such that a and b are just positive numbers, and the major axis could be either. But the problem mentions the upper vertex and the right focus. So, the upper vertex is (0, b) and the right focus is (c, 0). So, that would imply that the ellipse has both x and y axes as axes of symmetry, but the major axis could be either. Wait, but if the upper vertex is (0, b) and the right focus is (c, 0), then the major axis must be along the y-axis because the upper vertex is a vertex, not a co-vertex. So, if the upper vertex is (0, b), then b must be greater than a, making it a vertical ellipse.So, in that case, c = sqrt(b² - a²), and the foci are at (0, ±c). But the problem mentions the right focus, which would be (c, 0). Wait, but if it's a vertical ellipse, the foci are along the y-axis, so (c, 0) wouldn't be a focus. Hmm, this is confusing.Wait, maybe I'm misinterpreting the problem. Maybe the ellipse is such that both the upper vertex and the right focus lie on the line 2x + y - 2 = 0. So, regardless of the orientation, the points (0, b) and (c, 0) lie on this line.So, let's try that approach. Let me denote the upper vertex as (0, b) and the right focus as (c, 0). Since both these points lie on the line 2x + y - 2 = 0, we can plug these points into the line equation.First, plugging in (0, b):2*0 + b - 2 = 0 ⇒ b - 2 = 0 ⇒ b = 2.Okay, so b is 2.Next, plugging in (c, 0):2*c + 0 - 2 = 0 ⇒ 2c - 2 = 0 ⇒ 2c = 2 ⇒ c = 1.So, c is 1.Now, for an ellipse, we have the relationship between a, b, and c. Depending on whether it's a horizontal or vertical ellipse, the relationship changes.If it's a horizontal ellipse (a > b), then c = sqrt(a² - b²).If it's a vertical ellipse (b > a), then c = sqrt(b² - a²).But in our case, we have c = 1, and b = 2. So, let's see:If it's a horizontal ellipse, then c = sqrt(a² - b²) ⇒ 1 = sqrt(a² - 4) ⇒ a² - 4 = 1 ⇒ a² = 5 ⇒ a = sqrt(5).But if it's a vertical ellipse, then c = sqrt(b² - a²) ⇒ 1 = sqrt(4 - a²) ⇒ 4 - a² = 1 ⇒ a² = 3 ⇒ a = sqrt(3).Wait, but the problem didn't specify whether a > b or b > a. Hmm. But in the standard form x²/a² + y²/b² = 1, usually a is associated with the x-axis and b with the y-axis. So, if a > b, it's horizontal; if b > a, it's vertical.But in our case, we have b = 2 and c = 1. If it's a horizontal ellipse, then a² = b² + c² = 4 + 1 = 5, so a = sqrt(5). If it's a vertical ellipse, then a² = b² - c² = 4 - 1 = 3, so a = sqrt(3).But wait, in a vertical ellipse, the foci are along the y-axis, so the right focus wouldn't exist. Similarly, in a horizontal ellipse, the upper vertex is (0, b), which is a co-vertex, not a vertex. So, perhaps the problem is considering the upper vertex as (0, b) regardless of the orientation, and the right focus as (c, 0). So, perhaps the ellipse is horizontal, with a > b, and the upper vertex is (0, b), which is a co-vertex, and the right focus is (c, 0).Wait, but in that case, the upper vertex is not a vertex but a co-vertex. So, maybe the problem is using "upper vertex" to mean the topmost point, which would be (0, b) if it's a vertical ellipse, but in that case, the right focus wouldn't be (c, 0). Hmm, this is confusing.Wait, perhaps the problem is considering the ellipse to be horizontal, with a > b, so the vertices are at (±a, 0), and the co-vertices at (0, ±b). Then, the upper co-vertex is (0, b), and the right focus is (c, 0), where c = sqrt(a² - b²). So, in this case, both (0, b) and (c, 0) lie on the line 2x + y - 2 = 0.So, from (0, b): 2*0 + b - 2 = 0 ⇒ b = 2.From (c, 0): 2*c + 0 - 2 = 0 ⇒ c = 1.Then, since it's a horizontal ellipse, c = sqrt(a² - b²) ⇒ 1 = sqrt(a² - 4) ⇒ a² - 4 = 1 ⇒ a² = 5 ⇒ a = sqrt(5).Therefore, the equation of the ellipse is x²/5 + y²/4 = 1.Wait, but let me double-check. If it's a horizontal ellipse, then the major axis is along the x-axis, so the vertices are at (±sqrt(5), 0), and the co-vertices at (0, ±2). The foci are at (±1, 0). So, the right focus is (1, 0), and the upper co-vertex is (0, 2). Both these points lie on the line 2x + y - 2 = 0.Plugging (0, 2): 2*0 + 2 - 2 = 0 ⇒ 0 = 0, which works.Plugging (1, 0): 2*1 + 0 - 2 = 0 ⇒ 0 = 0, which also works.So, that seems consistent. Therefore, the equation of the ellipse is x²/5 + y²/4 = 1.I think that's the answer.

💡Alright, so I've got this problem here about a piecewise function, and I need to find the range of values for a real number ( a ) such that ( f(a) > f(-a) + 2 ). The function ( f(x) ) is defined differently depending on whether ( x ) is positive or negative. Let me try to break this down step by step.First, let's write down the function clearly:[f(x) = begin{cases}log_{2}x & text{if } x > 0 log_{frac{1}{2}}(-x) & text{if } x < 0end{cases}]Okay, so for positive ( x ), it's a logarithm with base 2, and for negative ( x ), it's a logarithm with base ( frac{1}{2} ) of the negative of ( x ). That makes sense because the logarithm function isn't defined for non-positive numbers, so they're adjusting it for negative inputs.Now, the inequality we need to solve is ( f(a) > f(-a) + 2 ). Since ( a ) can be positive or negative, I think I need to consider two separate cases: one where ( a > 0 ) and another where ( a < 0 ). Let's tackle each case one by one.**Case 1: ( a > 0 )**If ( a ) is positive, then ( f(a) ) will use the first part of the piecewise function:[f(a) = log_{2}a]Now, what about ( f(-a) )? Since ( a ) is positive, ( -a ) is negative, so we'll use the second part of the function:[f(-a) = log_{frac{1}{2}}(-(-a)) = log_{frac{1}{2}}a]Wait, hold on. The function for ( x < 0 ) is ( log_{frac{1}{2}}(-x) ), so substituting ( x = -a ), we get ( log_{frac{1}{2}}(-(-a)) = log_{frac{1}{2}}a ). That seems correct.So, plugging these into the inequality:[log_{2}a > log_{frac{1}{2}}a + 2]Hmm, I need to solve this inequality for ( a ). Let me recall that ( log_{frac{1}{2}}a ) can be rewritten using the change of base formula or properties of logarithms. Since ( log_{frac{1}{2}}a = frac{log_{2}a}{log_{2}frac{1}{2}} ). And ( log_{2}frac{1}{2} = -1 ), so:[log_{frac{1}{2}}a = frac{log_{2}a}{-1} = -log_{2}a]So, substituting back into the inequality:[log_{2}a > -log_{2}a + 2]Let's bring all the terms involving ( log_{2}a ) to one side:[log_{2}a + log_{2}a > 2][2log_{2}a > 2]Divide both sides by 2:[log_{2}a > 1]Now, solving for ( a ):[a > 2^{1} = 2]So, for ( a > 0 ), the inequality holds when ( a > 2 ).**Case 2: ( a < 0 )**Now, let's consider when ( a ) is negative. In this case, ( f(a) ) will use the second part of the piecewise function:[f(a) = log_{frac{1}{2}}(-a)]And ( f(-a) ) will use the first part because ( -a ) is positive:[f(-a) = log_{2}(-a)]Wait, let me double-check that. If ( a ) is negative, then ( -a ) is positive, so yes, ( f(-a) = log_{2}(-a) ).So, plugging these into the inequality:[log_{frac{1}{2}}(-a) > log_{2}(-a) + 2]Again, I can use the change of base formula to express ( log_{frac{1}{2}}(-a) ) in terms of base 2. Let's do that:[log_{frac{1}{2}}(-a) = frac{log_{2}(-a)}{log_{2}frac{1}{2}} = frac{log_{2}(-a)}{-1} = -log_{2}(-a)]So, substituting back into the inequality:[-log_{2}(-a) > log_{2}(-a) + 2]Let me bring all the ( log_{2}(-a) ) terms to one side:[-log_{2}(-a) - log_{2}(-a) > 2][-2log_{2}(-a) > 2]Divide both sides by -2. But wait, when I divide or multiply both sides of an inequality by a negative number, the inequality sign flips. So:[log_{2}(-a) < -1]Now, solving for ( -a ):[log_{2}(-a) < -1 implies -a < 2^{-1} = frac{1}{2}]Since ( a ) is negative, ( -a ) is positive, so:[-a < frac{1}{2} implies a > -frac{1}{2}]But remember, in this case, ( a < 0 ). So combining these two results, we have:[-frac{1}{2} < a < 0]**Combining Both Cases**From Case 1, we have ( a > 2 ).From Case 2, we have ( -frac{1}{2} < a < 0 ).So, the solution set is the union of these two intervals:[left( -frac{1}{2}, 0 right) cup left( 2, +infty right)]Looking back at the options provided:A: ( left( -frac{1}{2},0 right)cup left( 0,2 right) )B: ( left( -infty ,-frac{1}{2} right)cup left( 2,+infty right) )C: ( left( -frac{1}{2},0 right)cup left( 2,+infty right) )D: ( left( -infty ,-frac{1}{2} right)cup left( 0,2 right) )So, the correct answer is option C.**Final Answer**boxed{C}