💡First, I need to find the slope of the line passing through the points (7, 3) and (3, -1). The slope formula is m = (y2 - y1)/(x2 - x1). Plugging in the values, I get m = (-1 - 3)/(3 - 7) = (-4)/(-4) = 1.Next, I'll use the point-slope form of the equation of a line, which is y - y1 = m(x - x1). Using the point (7, 3), the equation becomes y - 3 = 1(x - 7). Simplifying this, I get y = x - 4.To find where this line intersects the x-axis, I set y = 0. So, 0 = x - 4. Solving for x, I find x = 4. Therefore, the line intersects the x-axis at the point (4, 0).

💡Okay, so I have this problem where I need to place 4 identical chess pieces on a fixed 3x3 grid. The rules are that no more than one piece can be placed in any single cell, and each row and each column must contain at least one piece. I need to figure out how many different ways this can be done.Hmm, let me start by understanding the problem better. A 3x3 grid has 9 cells, and I need to place 4 pieces such that every row and every column has at least one piece. That means I can't have any row or column completely empty. Since the pieces are identical, the order in which I place them doesn't matter, only their positions.First, I should think about how to distribute the 4 pieces across the 3 rows. Since there are 4 pieces and 3 rows, by the pigeonhole principle, at least one row must have 2 pieces, and the other two rows will have 1 piece each. Similarly, for the columns, since there are 4 pieces and 3 columns, at least one column must have 2 pieces, and the other two columns will have 1 piece each.So, my task is to count the number of ways to arrange the pieces such that each row and each column has at least one piece. This seems like a combinatorial problem where I need to consider both row and column constraints.Let me try to approach this step by step.**Step 1: Choosing the Row with 2 Pieces**There are 3 rows in the grid. I need to choose which row will have 2 pieces. There are 3 choices for this.**Step 2: Choosing Positions in the Selected Row**Once I've chosen the row that will have 2 pieces, I need to choose 2 cells out of the 3 in that row. The number of ways to choose 2 cells from 3 is given by the combination formula:[binom{3}{2} = 3]So, for each chosen row, there are 3 ways to place the 2 pieces.**Step 3: Placing the Remaining 2 Pieces in the Other Rows**Now, I have 2 pieces left to place in the remaining 2 rows. Each of these rows must have exactly 1 piece, and each column must also have at least 1 piece.This is where it gets a bit tricky because I have to ensure that the placement of these remaining pieces doesn't violate the column constraints.Let me visualize the grid. Suppose I've already placed 2 pieces in the first row. Now, I need to place 1 piece each in the second and third rows. Each of these pieces must go into different columns to ensure that each column has at least one piece.Wait, but since I've already placed 2 pieces in the first row, those pieces are in two different columns. So, the remaining two columns (since there are 3 columns) must each have at least one piece in the remaining two rows.But hold on, I have 2 pieces left and 2 rows left. Each of these pieces must go into different columns. So, essentially, I need to place one piece in each of the remaining two columns in the second and third rows.But how many ways can I do this?Let me think of it as assigning the remaining two pieces to the remaining two columns. Since the pieces are identical, the order doesn't matter, but the positions do.Wait, no, actually, since the pieces are identical, but the grid positions are distinct, the number of ways to assign the pieces to the columns is equal to the number of ways to assign the columns to the rows.So, for the second row, I can choose any of the 3 columns, but I have to make sure that the third row gets a different column.But actually, since I've already placed 2 pieces in the first row, which occupy 2 columns, the remaining column is the one that hasn't been used yet. So, the second and third rows must each have a piece in the remaining column and one of the other columns.Wait, no, that might not necessarily be the case. Let me think again.If I've placed 2 pieces in the first row, say in columns 1 and 2, then columns 1 and 2 already have pieces. The third column is still empty. So, the remaining two pieces must be placed in the second and third rows, but they can be in any columns, but we have to ensure that each column has at least one piece.So, the third column must have at least one piece. Therefore, one of the remaining two pieces must be placed in the third column. The other piece can be placed in either column 1 or 2.But since the first row already has pieces in columns 1 and 2, placing another piece in column 1 or 2 would just add to those columns, but they already have pieces, so it's allowed.Wait, but the pieces are identical, so it doesn't matter which specific piece goes where, just the positions.So, for the second row, I can choose any column, but for the third row, I have to choose a different column if necessary.Wait, maybe it's better to think in terms of permutations.Since I have two rows (second and third) and three columns, but two columns are already occupied by the first row. So, I need to place one piece in each of the remaining two rows, such that each column has at least one piece.So, the third column must have at least one piece, so one of the remaining two pieces must be in the third column. The other piece can be in either of the first two columns.So, for the second row, I can choose column 3, and then the third row can choose either column 1 or 2. Alternatively, the second row can choose column 1 or 2, and the third row must choose column 3.So, the number of ways is:- Choose which row (second or third) gets column 3: 2 choices.- For the other row, choose between the remaining two columns: 2 choices.So, total ways: 2 * 2 = 4.But wait, is that correct? Let me think.If I fix the second row to column 3, then the third row can be in column 1 or 2: 2 ways.Similarly, if I fix the third row to column 3, the second row can be in column 1 or 2: 2 ways.So, total 4 ways.But wait, is that all? Or are there more possibilities?Wait, no, because the first row already has two pieces, so the remaining columns are only column 3, and columns 1 and 2 are already occupied. So, the remaining two pieces must be placed in column 3 and one of columns 1 or 2.Therefore, the number of ways is 2 (choices for which row gets column 3) multiplied by 2 (choices for the other column), giving 4 ways.But hold on, is that accurate? Let me think of it as assigning the columns to the rows.We have two rows (second and third) and three columns, but two columns are already occupied by the first row. So, we need to assign columns to these two rows such that column 3 is assigned to at least one of them.So, the number of ways is equal to the number of ways to assign columns to the two rows, with the constraint that column 3 is assigned to at least one row.The total number of ways without any constraints is 3 * 3 = 9 (each row can choose any column). But we need to subtract the cases where column 3 is not chosen at all.The number of ways where column 3 is not chosen is 2 * 2 = 4 (each row chooses between columns 1 and 2).Therefore, the number of valid ways is 9 - 4 = 5.Wait, that contradicts my earlier conclusion of 4 ways. Which one is correct?Let me think again.If I have two rows and three columns, with the constraint that column 3 must be chosen at least once.Total assignments: 3^2 = 9.Subtract the assignments where column 3 is never chosen: 2^2 = 4.So, valid assignments: 9 - 4 = 5.Therefore, there are 5 ways to assign the remaining two pieces to the second and third rows such that column 3 is occupied.But how does this align with my previous reasoning?Earlier, I thought that one row must be assigned to column 3, and the other row can be assigned to either column 1 or 2, giving 2 * 2 = 4 ways. But according to this, it's 5 ways.Hmm, so which one is correct?Wait, perhaps I made a mistake in my initial reasoning.Let me list all possible assignments:1. Second row: column 1, Third row: column 32. Second row: column 2, Third row: column 33. Second row: column 3, Third row: column 14. Second row: column 3, Third row: column 25. Second row: column 3, Third row: column 3Wait, hold on, can both the second and third rows be assigned to column 3? That would mean two pieces in column 3, which is allowed because we have 4 pieces in total, and column 3 can have up to 2 pieces.But in this case, since we have only 2 pieces left, assigning both to column 3 would mean column 3 has 2 pieces, and the other columns have 1 each. That's acceptable because each column must have at least one piece.So, in fact, there are 5 valid assignments:1. Second row: 1, Third row: 32. Second row: 2, Third row: 33. Second row: 3, Third row: 14. Second row: 3, Third row: 25. Second row: 3, Third row: 3Therefore, there are 5 ways, not 4. So, my initial reasoning was incorrect because I didn't consider the case where both remaining pieces are placed in column 3.Therefore, the number of ways to place the remaining two pieces is 5.So, putting it all together:- Choose which row has 2 pieces: 3 ways.- Choose 2 cells in that row: 3 ways.- Place the remaining 2 pieces: 5 ways.Total number of ways: 3 * 3 * 5 = 45.Wait, but let me verify this with another approach to make sure.**Alternative Approach: Using Inclusion-Exclusion**Another way to approach this problem is to calculate the total number of ways to place 4 pieces on the 3x3 grid without any constraints and then subtract the configurations that violate the row or column constraints.Total number of ways to place 4 identical pieces on 9 cells:[binom{9}{4} = 126]Now, we need to subtract the configurations where at least one row is empty and/or at least one column is empty.But this can get complicated because we have to consider overlapping cases where both rows and columns are empty. This is where the inclusion-exclusion principle comes into play.Let me try to outline the steps:1. Calculate the total number of ways without constraints: 126.2. Subtract the configurations where at least one row is empty. - Number of ways where a specific row is empty: Choose 4 cells from the remaining 6 cells (since one row is empty). There are 3 rows, so: [ 3 times binom{6}{4} = 3 times 15 = 45 ]3. Similarly, subtract the configurations where at least one column is empty. - Number of ways where a specific column is empty: Choose 4 cells from the remaining 6 cells (since one column is empty). There are 3 columns, so: [ 3 times binom{6}{4} = 3 times 15 = 45 ]4. However, by subtracting both row and column constraints, we have subtracted too much because some configurations were subtracted twice (those where both a row and a column are empty). So, we need to add those back. - Number of ways where a specific row and a specific column are both empty: Choose 4 cells from the remaining 4 cells (since one row and one column are empty). There are 3 rows and 3 columns, so 9 such combinations: [ 9 times binom{4}{4} = 9 times 1 = 9 ]5. Now, applying the inclusion-exclusion principle: Total valid configurations = Total configurations - configurations with empty rows - configurations with empty columns + configurations with both empty rows and columns [ 126 - 45 - 45 + 9 = 45 ]So, this method also gives us 45 valid configurations.Therefore, both methods lead to the same result: 45 ways.I think this confirms that the initial calculation was correct, and the answer is indeed 45.**Final Answer**The number of different ways to place the 4 identical chess pieces is boxed{45}.

💡First, I recognize that the coin has two possible outcomes for each toss: heads (H) or tails (T), each with an equal probability of 0.5.To find the probability of the coin landing on the same side every time when tossed 4 times, I need to consider two scenarios:1. Getting heads on all 4 tosses (HHHH)2. Getting tails on all 4 tosses (TTTT)For each scenario, the probability is calculated by multiplying the probability of each individual outcome:- Probability of HHHH = 0.5 * 0.5 * 0.5 * 0.5 = 0.0625- Probability of TTTT = 0.5 * 0.5 * 0.5 * 0.5 = 0.0625Since these two events (all heads or all tails) are mutually exclusive, I can add their probabilities to find the total probability of the coin landing on the same side each time:0.0625 + 0.0625 = 0.125Therefore, the probability that the coin will land on the same side each time when tossed 4 times is 0.125 or 12.5%.

💡Okay, so I have this problem with four inequalities, and I need to figure out which one holds true for any real number x. Let me go through each option one by one and see if I can determine which one is always true.Starting with option A: (frac{1}{x^2 + 1} < 1). Hmm, let me think about this. The denominator here is (x^2 + 1), which is always positive because (x^2) is non-negative and adding 1 makes it at least 1. So, the fraction (frac{1}{x^2 + 1}) will always be positive. But is it always less than 1?Well, let's test some values. If x is 0, then the expression becomes (frac{1}{0 + 1} = 1). So, 1 is not less than 1; it's equal. That means when x is 0, the inequality doesn't hold. Therefore, option A isn't true for all real numbers x. So, I can eliminate option A.Moving on to option B: (x^2 + 1 geqslant 2|x|). This looks a bit more promising. Let me try to analyze this inequality. Maybe I can rearrange it or use some known inequality.I remember that the expression (x^2 + 1) can be related to the square of something. Let me consider (x^2 - 2|x| + 1). That looks familiar—it's similar to the expansion of ((|x| - 1)^2), which is (|x|^2 - 2|x| + 1). Since (|x|^2 = x^2), this is exactly the same as (x^2 - 2|x| + 1).So, (x^2 + 1 - 2|x| = (|x| - 1)^2). Since a square is always non-negative, ((|x| - 1)^2 geq 0) for all real x. Therefore, (x^2 + 1 geqslant 2|x|) must be true for all real numbers x. That seems solid. I'll keep option B as a possible answer.Now, option C: (lg(x^2 + 1) geqslant lg 2x). Hmm, logarithms can be tricky. First, I should remember that the logarithm function is only defined for positive arguments. So, for (lg 2x) to be defined, (2x) must be positive, which means x must be greater than 0. But the problem says "for any (x in mathbb{R})", which includes negative numbers and zero. If x is negative, (lg 2x) isn't defined, so the inequality doesn't hold for negative x. Also, if x is 0, (lg 2x) is undefined as well. Therefore, option C isn't valid for all real numbers. So, I can rule out option C.Lastly, option D: (frac{4x}{x^2 + 4} geqslant 1). Let me see. Again, let's test some values. If x is positive, maybe it works, but what about negative x?Let's try x = 0. Plugging in, we get (frac{0}{0 + 4} = 0), which is not greater than or equal to 1. So, already, when x is 0, the inequality doesn't hold. Therefore, option D isn't true for all real numbers.Wait, but maybe I should check another value. Let's try x = 2. Then, (frac{4*2}{4 + 4} = frac{8}{8} = 1), which is equal to 1, so that's okay. What about x = 1? (frac{4*1}{1 + 4} = frac{4}{5} = 0.8), which is less than 1. So, that doesn't satisfy the inequality either. So, even for positive x, it doesn't always hold. So, definitely, option D isn't the right answer.So, after going through all the options, only option B holds true for any real number x. I think that's the answer.**Final Answer**boxed{B}

💡First, I need to determine the length of the train. The train is moving at a speed of 108 kilometers per hour and takes 7 seconds to pass a pole completely.To calculate the length of the train, I should convert the speed from kilometers per hour to meters per second. This conversion is necessary because the time is given in seconds and the desired unit for the length is meters.The conversion formula is:[text{Speed in m/s} = text{Speed in km/hr} times frac{1000 text{ meters}}{1 text{ kilometer}} times frac{1 text{ hour}}{3600 text{ seconds}}]Applying the values:[text{Speed in m/s} = 108 times frac{1000}{3600} = 108 times frac{5}{18} = 30 text{ m/s}]Now that the speed is in meters per second, I can calculate the length of the train using the formula:[text{Distance} = text{Speed} times text{Time}]Substituting the known values:[text{Length of the train} = 30 text{ m/s} times 7 text{ s} = 210 text{ meters}]Therefore, the length of the train is 210 meters.

💡Okay, so I have this problem where I need to find the maximum value of the expression |(z + i)/(z + 2)| where z is a complex number with modulus 1. Hmm, let me try to figure this out step by step.First, I know that if |z| = 1, then z lies on the unit circle in the complex plane. So z can be represented as z = cosθ + i sinθ, where θ is the angle made with the positive real axis. That might be useful.Let me write down the expression I need to maximize:|(z + i)/(z + 2)|Since z is on the unit circle, let's substitute z = cosθ + i sinθ into the expression.So, substituting z:Numerator: z + i = cosθ + i sinθ + i = cosθ + i(sinθ + 1)Denominator: z + 2 = cosθ + 2 + i sinθNow, the modulus of a complex number a + ib is sqrt(a² + b²). So, let's compute the modulus of the numerator and denominator separately.First, modulus of numerator:|z + i| = sqrt[ (cosθ)² + (sinθ + 1)² ]Let me expand that:= sqrt[ cos²θ + (sin²θ + 2 sinθ + 1) ]= sqrt[ cos²θ + sin²θ + 2 sinθ + 1 ]Since cos²θ + sin²θ = 1, this simplifies to:= sqrt[ 1 + 2 sinθ + 1 ]= sqrt[ 2 + 2 sinθ ]= sqrt[2(1 + sinθ)]Okay, so |z + i| = sqrt(2(1 + sinθ))Now, modulus of denominator:|z + 2| = sqrt[ (cosθ + 2)² + (sinθ)² ]Expanding that:= sqrt[ cos²θ + 4 cosθ + 4 + sin²θ ]Again, cos²θ + sin²θ = 1, so:= sqrt[ 1 + 4 cosθ + 4 ]= sqrt[5 + 4 cosθ]So, |z + 2| = sqrt(5 + 4 cosθ)Therefore, the expression we're trying to maximize is:| (z + i)/(z + 2) | = |z + i| / |z + 2| = sqrt(2(1 + sinθ)) / sqrt(5 + 4 cosθ)Let me denote this as:f(θ) = sqrt(2(1 + sinθ)) / sqrt(5 + 4 cosθ)To find the maximum value of f(θ), it's equivalent to maximizing f(θ)², since the square root function is monotonically increasing.So, let's compute f(θ)²:f(θ)² = [2(1 + sinθ)] / [5 + 4 cosθ]Let me denote this as p(θ) = [2(1 + sinθ)] / [5 + 4 cosθ]So, we need to find the maximum of p(θ).To maximize p(θ), we can use calculus. Let's consider p(θ) as a function of θ and find its critical points.Alternatively, perhaps we can use some trigonometric identities or inequalities to find the maximum.Let me try to express p(θ) in terms of sine and cosine and see if I can manipulate it.So, p(θ) = [2(1 + sinθ)] / [5 + 4 cosθ]Let me write this as:p(θ) = (2 + 2 sinθ) / (5 + 4 cosθ)Hmm, perhaps I can write this as:p(θ) = (2 sinθ + 2) / (4 cosθ + 5)This looks like a linear combination of sinθ and cosθ in both numerator and denominator.I remember that expressions of the form A sinθ + B cosθ can be rewritten using the amplitude-phase form, but I'm not sure if that's directly applicable here.Alternatively, maybe I can use the method of Lagrange multipliers or consider this as an optimization problem with constraints.Wait, another approach is to consider this as a function of θ and take its derivative with respect to θ, set the derivative equal to zero, and solve for θ.Let me try that.Let me denote:p(θ) = (2 + 2 sinθ) / (5 + 4 cosθ)Let me compute dp/dθ.Using the quotient rule:dp/dθ = [ (2 cosθ)(5 + 4 cosθ) - (2 + 2 sinθ)(-4 sinθ) ] / (5 + 4 cosθ)^2Simplify numerator:= [10 cosθ + 8 cos²θ + 8 sinθ + 8 sin²θ] / (5 + 4 cosθ)^2Wait, let me double-check that.Wait, the derivative of the numerator is 2 cosθ, and the derivative of the denominator is -4 sinθ.So, using the quotient rule:dp/dθ = [ (2 cosθ)(5 + 4 cosθ) - (2 + 2 sinθ)(-4 sinθ) ] / (5 + 4 cosθ)^2Compute numerator:First term: 2 cosθ * (5 + 4 cosθ) = 10 cosθ + 8 cos²θSecond term: - (2 + 2 sinθ) * (-4 sinθ) = (2 + 2 sinθ)(4 sinθ) = 8 sinθ + 8 sin²θSo total numerator:10 cosθ + 8 cos²θ + 8 sinθ + 8 sin²θSo, dp/dθ = [10 cosθ + 8 cos²θ + 8 sinθ + 8 sin²θ] / (5 + 4 cosθ)^2To find critical points, set numerator equal to zero:10 cosθ + 8 cos²θ + 8 sinθ + 8 sin²θ = 0Hmm, this seems complicated. Maybe I can factor or simplify this expression.Let me note that 8 cos²θ + 8 sin²θ = 8 (cos²θ + sin²θ) = 8 * 1 = 8So, the equation becomes:10 cosθ + 8 + 8 sinθ = 0So,10 cosθ + 8 sinθ + 8 = 0Let me write this as:10 cosθ + 8 sinθ = -8Hmm, this is a linear combination of sinθ and cosθ. Maybe I can write this as R cos(θ - φ) = -8, where R is the amplitude.Compute R:R = sqrt(10² + 8²) = sqrt(100 + 64) = sqrt(164) = 2 sqrt(41)So,10 cosθ + 8 sinθ = R cos(θ - φ) = 2 sqrt(41) cos(θ - φ) = -8So,cos(θ - φ) = -8 / (2 sqrt(41)) = -4 / sqrt(41)So,θ - φ = arccos(-4 / sqrt(41)) or θ - φ = -arccos(-4 / sqrt(41))So,θ = φ ± arccos(-4 / sqrt(41))But I need to find φ such that:cosφ = 10 / R = 10 / (2 sqrt(41)) = 5 / sqrt(41)sinφ = 8 / R = 8 / (2 sqrt(41)) = 4 / sqrt(41)So, φ = arctan(8/10) = arctan(4/5)So, θ = arctan(4/5) ± arccos(-4 / sqrt(41))Hmm, this is getting a bit involved. Maybe I can compute the value numerically, but perhaps there's a better approach.Alternatively, maybe I can use the method of expressing the ratio as a function and find its maximum.Wait, another approach is to consider the expression p(θ) = (2 + 2 sinθ)/(5 + 4 cosθ) and try to maximize it.Let me denote x = sinθ and y = cosθ, with the constraint that x² + y² = 1.So, p = (2 + 2x)/(5 + 4y)We need to maximize p subject to x² + y² = 1.This is a constrained optimization problem. I can use Lagrange multipliers.Let me set up the Lagrangian:L = (2 + 2x)/(5 + 4y) + λ(x² + y² - 1)Wait, actually, since p is a function to be maximized, perhaps it's better to set up the derivative conditions.Alternatively, maybe I can parametrize x and y in terms of t, but perhaps another substitution.Alternatively, maybe I can write this as:Let me denote t = tan(θ/2), which is the Weierstrass substitution.But that might complicate things.Alternatively, perhaps I can consider p as a function and write it in terms of y.From x² + y² = 1, x = sqrt(1 - y²) or x = -sqrt(1 - y²). But since we're maximizing, perhaps we can consider x positive.Wait, but in our expression, p = (2 + 2x)/(5 + 4y), so to maximize p, we need to maximize the numerator and minimize the denominator.But since x and y are related by x² + y² = 1, perhaps we can express x in terms of y.Let me try that.Let me write x = sqrt(1 - y²). Then,p = (2 + 2 sqrt(1 - y²))/(5 + 4y)We can consider this as a function of y, where y ∈ [-1, 1].Let me denote this as p(y) = (2 + 2 sqrt(1 - y²))/(5 + 4y)To find the maximum, take derivative of p with respect to y and set to zero.Compute dp/dy:Let me denote numerator as N = 2 + 2 sqrt(1 - y²)Denominator as D = 5 + 4ySo, dp/dy = (N’ D - N D’) / D²Compute N’:N’ = 2 * (1/(2 sqrt(1 - y²))) * (-2y) = (-2y)/sqrt(1 - y²)Compute D’:D’ = 4So,dp/dy = [ (-2y)/sqrt(1 - y²) * (5 + 4y) - (2 + 2 sqrt(1 - y²)) * 4 ] / (5 + 4y)^2Set dp/dy = 0, so numerator must be zero:(-2y)(5 + 4y)/sqrt(1 - y²) - 4(2 + 2 sqrt(1 - y²)) = 0This seems complicated. Maybe I can multiply both sides by sqrt(1 - y²) to eliminate the denominator:-2y(5 + 4y) - 4(2 + 2 sqrt(1 - y²)) sqrt(1 - y²) = 0Wait, let me check:Wait, the original numerator is:(-2y)(5 + 4y)/sqrt(1 - y²) - 4(2 + 2 sqrt(1 - y²)) = 0Multiplying both sides by sqrt(1 - y²):-2y(5 + 4y) - 4(2 + 2 sqrt(1 - y²)) sqrt(1 - y²) = 0Wait, that might not be the best approach. Maybe there's a better substitution.Alternatively, perhaps I can consider the expression p(θ) and use the method of expressing it in terms of a single trigonometric function.Wait, another approach is to consider the expression p(θ) = (2 + 2 sinθ)/(5 + 4 cosθ) and use the Cauchy-Schwarz inequality or some other inequality to bound it.Alternatively, perhaps I can write this as:p(θ) = [2(1 + sinθ)] / [5 + 4 cosθ]Let me consider 1 + sinθ. I know that 1 + sinθ can be written as 2 sin²(θ/2 + π/4) or something similar, but maybe that's not helpful.Alternatively, perhaps I can write 1 + sinθ = 2 sin(θ/2 + π/4)^2, but I'm not sure.Wait, another idea: Let me consider the expression as a function of θ and try to find its maximum by considering it as a function of a single variable.Alternatively, perhaps I can use the method of expressing the ratio as a function and find its maximum by considering it as a function of θ.Wait, maybe I can consider the expression p(θ) = (2 + 2 sinθ)/(5 + 4 cosθ) and try to find its maximum by considering the derivative.But earlier, when I tried to compute the derivative, it got complicated. Maybe I can try to solve the equation 10 cosθ + 8 sinθ = -8.Let me write this as:10 cosθ + 8 sinθ = -8Let me divide both sides by 2:5 cosθ + 4 sinθ = -4Let me write this as:5 cosθ + 4 sinθ = -4I can write this as R cos(θ - φ) = -4, where R = sqrt(5² + 4²) = sqrt(25 + 16) = sqrt(41)So,sqrt(41) cos(θ - φ) = -4Thus,cos(θ - φ) = -4 / sqrt(41)So,θ - φ = arccos(-4 / sqrt(41)) or θ - φ = -arccos(-4 / sqrt(41))So,θ = φ ± arccos(-4 / sqrt(41))Where φ is such that:cosφ = 5 / sqrt(41)sinφ = 4 / sqrt(41)So, φ = arctan(4/5)Thus,θ = arctan(4/5) ± arccos(-4 / sqrt(41))Hmm, this seems a bit involved, but perhaps I can compute the value of θ.Alternatively, perhaps I can find the maximum value without explicitly finding θ.Wait, another approach: Let me consider the expression p(θ) = (2 + 2 sinθ)/(5 + 4 cosθ)Let me denote this as p = (2(1 + sinθ))/(5 + 4 cosθ)Let me consider this as a function of θ and try to find its maximum.Let me use the method of expressing this as a function and finding its maximum by considering the derivative.Wait, perhaps I can use the substitution t = tan(θ/2), which is the Weierstrass substitution.Let me set t = tan(θ/2), so that sinθ = 2t/(1 + t²) and cosθ = (1 - t²)/(1 + t²)Then, p becomes:p = [2(1 + 2t/(1 + t²))]/[5 + 4*(1 - t²)/(1 + t²)]Simplify numerator:= [2*( (1 + t²) + 2t ) / (1 + t²) ]= [2*(1 + t² + 2t)] / (1 + t²)= [2*(t + 1)^2] / (1 + t²)Denominator:= [5*(1 + t²) + 4*(1 - t²)] / (1 + t²)= [5 + 5t² + 4 - 4t²] / (1 + t²)= [9 + t²] / (1 + t²)So, p = [2*(t + 1)^2 / (1 + t²)] / [ (9 + t²)/(1 + t²) ) ]= [2*(t + 1)^2 / (1 + t²)] * [ (1 + t²)/(9 + t²) ) ]= 2*(t + 1)^2 / (9 + t²)So, p = 2*(t + 1)^2 / (t² + 9)Now, we need to maximize p with respect to t, where t is real (since θ is real, t can be any real number).So, let me denote f(t) = 2*(t + 1)^2 / (t² + 9)To find the maximum of f(t), take its derivative and set it to zero.Compute f'(t):f'(t) = [2*2*(t + 1)*(1)*(t² + 9) - 2*(t + 1)^2*(2t)] / (t² + 9)^2Wait, let me compute it step by step.f(t) = 2*(t + 1)^2 / (t² + 9)Let me denote numerator as N = 2*(t + 1)^2 and denominator as D = t² + 9Then,f'(t) = (N’ D - N D’) / D²Compute N’:N = 2*(t + 1)^2N’ = 2*2*(t + 1) = 4(t + 1)Compute D’:D = t² + 9D’ = 2tSo,f'(t) = [4(t + 1)(t² + 9) - 2*(t + 1)^2*(2t)] / (t² + 9)^2Simplify numerator:= 4(t + 1)(t² + 9) - 4t(t + 1)^2Factor out 4(t + 1):= 4(t + 1)[(t² + 9) - t(t + 1)]Simplify inside the brackets:= (t² + 9) - t(t + 1) = t² + 9 - t² - t = 9 - tSo, numerator becomes:= 4(t + 1)(9 - t)Thus,f'(t) = [4(t + 1)(9 - t)] / (t² + 9)^2Set f'(t) = 0:4(t + 1)(9 - t) = 0So, solutions are t = -1 and t = 9Now, we need to check these critical points to see which gives the maximum.Compute f(t) at t = -1:f(-1) = 2*(-1 + 1)^2 / ((-1)^2 + 9) = 2*(0)^2 / (1 + 9) = 0/10 = 0Compute f(t) at t = 9:f(9) = 2*(9 + 1)^2 / (9² + 9) = 2*(100) / (81 + 9) = 200 / 90 = 20/9 ≈ 2.222Now, check the behavior as t approaches infinity:As t → ∞, f(t) ≈ 2*t² / t² = 2, so it approaches 2.Similarly, as t → -∞, f(t) approaches 2.So, the maximum occurs at t = 9, giving f(t) = 20/9.Therefore, the maximum value of p(θ) is 20/9, so the maximum value of |(z + i)/(z + 2)| is sqrt(20/9) = (2 sqrt(5))/3.Wait, let me check that:sqrt(20/9) = sqrt(20)/sqrt(9) = (2 sqrt(5))/3Yes, that's correct.So, the maximum value is (2 sqrt(5))/3.