💡Alright, so I have this problem here: "If each exterior angle of a polygon is 30 degrees, then the number of diagonals starting from a vertex of this polygon is: A: 7 diagonals, B: 8 diagonals, C: 9 diagonals, D: 10 diagonals." Hmm, okay, let me try to figure this out step by step.First, I remember that the sum of all exterior angles of any polygon is always 360 degrees. That's a key fact. So if each exterior angle is 30 degrees, I can find out how many sides the polygon has by dividing the total sum of exterior angles by the measure of each exterior angle. Let me write that down:Number of sides = Total sum of exterior angles / Each exterior angleNumber of sides = 360 degrees / 30 degreesNumber of sides = 12So, the polygon has 12 sides. That makes it a dodecagon, right? Okay, so now I know it's a 12-sided polygon.Next, I need to find the number of diagonals starting from a single vertex. I think there's a formula for that. Let me recall. I believe the number of diagonals from one vertex is equal to the total number of sides minus 3. Why minus 3? Well, because from any vertex, you can't draw a diagonal to itself, and you can't draw diagonals to the two adjacent vertices since those are connected by sides, not diagonals. So, it's n - 3, where n is the number of sides.Let me apply that formula:Number of diagonals from one vertex = n - 3Number of diagonals from one vertex = 12 - 3Number of diagonals from one vertex = 9So, that would mean there are 9 diagonals starting from each vertex of this 12-sided polygon. Let me double-check to make sure I didn't make a mistake. If each exterior angle is 30 degrees, then 360 divided by 30 is indeed 12 sides. And for a 12-sided polygon, subtracting 3 gives 9 diagonals per vertex. That seems right.Wait, just to be thorough, maybe I should visualize it or think about a simpler polygon to confirm the formula. Let's take a pentagon, which has 5 sides. Each vertex should have 5 - 3 = 2 diagonals. If I draw a pentagon, yes, from each vertex, there are two diagonals. Similarly, in a hexagon, which has 6 sides, each vertex should have 6 - 3 = 3 diagonals. Drawing a hexagon, that's correct too. So, the formula seems to hold.Therefore, applying it to a 12-sided polygon, it's 12 - 3 = 9 diagonals from each vertex. So, the answer should be C: 9 diagonals.Just to make sure, let me think about the total number of diagonals in the polygon. I remember there's a formula for the total number of diagonals in a polygon, which is n(n - 3)/2. For a 12-sided polygon, that would be 12*(12 - 3)/2 = 12*9/2 = 54. So, there are 54 diagonals in total. But the question is asking for the number starting from one vertex, which we've already determined is 9. So, that aligns with the total number as well.I think I'm confident now that the correct answer is C: 9 diagonals.

💡Alright, so I've got this problem about Ms. Daily Commuter who needs to get to her office by 9:00 AM. The problem says that if she drives at 30 mph, she'll be late by 5 minutes, and if she drives at 50 mph, she'll be early by 5 minutes. I need to figure out what speed she should drive to arrive exactly on time. The options are A) 35 mph, B) 37.5 mph, C) 40 mph, and D) 42 mph.Okay, let's break this down. First, I know that speed, time, and distance are related by the formula:[ text{Distance} = text{Speed} times text{Time} ]So, if I can figure out the distance to her office and the exact time she needs to get there, I can find the required speed.Let me define some variables to make this clearer. Let's say:- ( d ) is the distance to her office.- ( t ) is the time she needs to arrive exactly on time (in hours).Now, when she drives at 30 mph, she's late by 5 minutes. Since time is in hours, I should convert 5 minutes to hours. There are 60 minutes in an hour, so 5 minutes is ( frac{5}{60} ) hours, which simplifies to ( frac{1}{12} ) hours or approximately 0.0833 hours.Similarly, when she drives at 50 mph, she's early by 5 minutes, which is also ( frac{1}{12} ) hours.So, if she drives at 30 mph and is late, the time she takes is ( t + frac{1}{12} ) hours. And if she drives at 50 mph and is early, the time she takes is ( t - frac{1}{12} ) hours.Using the distance formula, I can write two equations:1. When driving at 30 mph:[ d = 30 times left( t + frac{1}{12} right) ]2. When driving at 50 mph:[ d = 50 times left( t - frac{1}{12} right) ]Since both expressions equal ( d ), I can set them equal to each other:[ 30 times left( t + frac{1}{12} right) = 50 times left( t - frac{1}{12} right) ]Now, I need to solve for ( t ). Let's expand both sides:Left side:[ 30 times t + 30 times frac{1}{12} = 30t + 2.5 ]Right side:[ 50 times t - 50 times frac{1}{12} = 50t - 4.1667 ]So, the equation becomes:[ 30t + 2.5 = 50t - 4.1667 ]Now, let's get all the terms involving ( t ) on one side and constants on the other:Subtract ( 30t ) from both sides:[ 2.5 = 20t - 4.1667 ]Add ( 4.1667 ) to both sides:[ 2.5 + 4.1667 = 20t ][ 6.6667 = 20t ]Now, divide both sides by 20:[ t = frac{6.6667}{20} ][ t = 0.3333 text{ hours} ]Hmm, 0.3333 hours. To convert that back to minutes, since 1 hour is 60 minutes, I multiply by 60:[ 0.3333 times 60 = 20 text{ minutes} ]So, the exact time she needs to take to arrive on time is 20 minutes.Now, let's find the distance ( d ). I'll use the first equation:[ d = 30 times left( t + frac{1}{12} right) ][ d = 30 times left( 0.3333 + 0.0833 right) ][ d = 30 times 0.4166 ][ d = 12.5 text{ miles} ]So, the distance to her office is 12.5 miles.Now, to find the speed she needs to drive to cover 12.5 miles in exactly 20 minutes, I'll use the speed formula:[ text{Speed} = frac{text{Distance}}{text{Time}} ]But first, I need to make sure the time is in hours. 20 minutes is ( frac{20}{60} ) hours, which simplifies to ( frac{1}{3} ) hours or approximately 0.3333 hours.So,[ text{Speed} = frac{12.5}{0.3333} ][ text{Speed} approx 37.5 text{ mph} ]Therefore, Ms. Daily Commuter needs to drive at approximately 37.5 mph to arrive exactly on time.Let me double-check my calculations to make sure I didn't make any mistakes.First, I set up the two equations correctly based on the given speeds and the time differences. Then, I solved for ( t ) and found it to be 0.3333 hours, which is 20 minutes. Then, I calculated the distance using the slower speed and got 12.5 miles. Finally, I used the distance and the exact time to find the required speed, which came out to be 37.5 mph.Everything seems to add up. So, I think 37.5 mph is the correct answer.

💡First, I need to determine the value of ( x ) that makes the average of the data set equal to 3. The average is calculated by summing all the data points and dividing by the number of points.Next, I'll calculate the sum of the known data points: ( 1 + 3 + 2 + 5 = 11 ). Since the average is 3 and there are 5 data points, the total sum should be ( 3 times 5 = 15 ). Therefore, ( x = 15 - 11 = 4 ).Now that I have all the data points, I'll calculate the variance. The variance is the average of the squared differences between each data point and the mean. I'll compute each squared difference:[(1 - 3)^2 = 4, quad (3 - 3)^2 = 0, quad (2 - 3)^2 = 1, quad (5 - 3)^2 = 4, quad (4 - 3)^2 = 1]Summing these squared differences gives ( 4 + 0 + 1 + 4 + 1 = 10 ). Dividing by the number of data points, the variance is ( frac{10}{5} = 2 ).Finally, the standard deviation is the square root of the variance, so ( sqrt{2} ).

💡Alright, so I've got this problem here: Given that ( a > 0 ), ( b > 0 ), and ( ab = 1 ), I need to find the minimum value of the expression ( frac{1}{2a} + frac{1}{2b} + frac{8}{a + b} ). Hmm, okay, let's see how to approach this.First off, since ( ab = 1 ), maybe I can express one variable in terms of the other. Let's say ( b = frac{1}{a} ). That way, I can rewrite the entire expression in terms of ( a ) only. Let me try that.So, substituting ( b = frac{1}{a} ) into the expression, I get:[frac{1}{2a} + frac{1}{2 cdot frac{1}{a}} + frac{8}{a + frac{1}{a}}]Simplifying each term:- The first term is ( frac{1}{2a} ).- The second term is ( frac{1}{2} cdot a ), because ( frac{1}{frac{1}{a}} = a ).- The third term is ( frac{8}{a + frac{1}{a}} ).So, the expression becomes:[frac{1}{2a} + frac{a}{2} + frac{8}{a + frac{1}{a}}]Hmm, that's a bit complicated. Maybe I can simplify the third term further. Let's see:[a + frac{1}{a} = frac{a^2 + 1}{a}]So, the third term becomes:[frac{8}{frac{a^2 + 1}{a}} = frac{8a}{a^2 + 1}]Now, the entire expression is:[frac{1}{2a} + frac{a}{2} + frac{8a}{a^2 + 1}]This still looks a bit messy. Maybe I can combine the first two terms:[frac{1}{2a} + frac{a}{2} = frac{1 + a^2}{2a}]So, now the expression is:[frac{1 + a^2}{2a} + frac{8a}{a^2 + 1}]Hmm, interesting. Let me denote ( t = a + frac{1}{a} ). Since ( a > 0 ), ( t geq 2 ) by AM-GM inequality, because ( a + frac{1}{a} geq 2sqrt{a cdot frac{1}{a}} = 2 ).But I'm not sure if that's helpful here. Maybe I can consider substituting ( t = a + frac{1}{a} ), but let's see:If ( t = a + frac{1}{a} ), then ( t^2 = a^2 + 2 + frac{1}{a^2} ), so ( a^2 + frac{1}{a^2} = t^2 - 2 ). Not sure if that helps.Alternatively, maybe I can consider the entire expression as a function of ( a ) and find its minimum by taking the derivative. That might be a bit involved, but let's try.Let me denote the expression as ( f(a) ):[f(a) = frac{1 + a^2}{2a} + frac{8a}{a^2 + 1}]Simplify ( f(a) ):[f(a) = frac{1}{2a} + frac{a}{2} + frac{8a}{a^2 + 1}]To find the minimum, I'll take the derivative ( f'(a) ) and set it equal to zero.First, compute the derivative term by term.1. Derivative of ( frac{1}{2a} ):[frac{d}{da} left( frac{1}{2a} right) = -frac{1}{2a^2}]2. Derivative of ( frac{a}{2} ):[frac{d}{da} left( frac{a}{2} right) = frac{1}{2}]3. Derivative of ( frac{8a}{a^2 + 1} ):Use the quotient rule: ( frac{d}{da} left( frac{u}{v} right) = frac{u'v - uv'}{v^2} ), where ( u = 8a ) and ( v = a^2 + 1 ).So,[u' = 8, quad v' = 2a]Thus,[frac{d}{da} left( frac{8a}{a^2 + 1} right) = frac{8(a^2 + 1) - 8a cdot 2a}{(a^2 + 1)^2} = frac{8a^2 + 8 - 16a^2}{(a^2 + 1)^2} = frac{-8a^2 + 8}{(a^2 + 1)^2} = frac{-8(a^2 - 1)}{(a^2 + 1)^2}]Putting it all together, the derivative ( f'(a) ) is:[f'(a) = -frac{1}{2a^2} + frac{1}{2} + frac{-8(a^2 - 1)}{(a^2 + 1)^2}]Simplify ( f'(a) ):[f'(a) = frac{1}{2} - frac{1}{2a^2} - frac{8(a^2 - 1)}{(a^2 + 1)^2}]To find critical points, set ( f'(a) = 0 ):[frac{1}{2} - frac{1}{2a^2} - frac{8(a^2 - 1)}{(a^2 + 1)^2} = 0]This equation looks quite complicated. Maybe I can multiply both sides by ( 2a^2(a^2 + 1)^2 ) to eliminate denominators:[2a^2(a^2 + 1)^2 cdot left( frac{1}{2} - frac{1}{2a^2} - frac{8(a^2 - 1)}{(a^2 + 1)^2} right) = 0]Simplify term by term:1. ( 2a^2(a^2 + 1)^2 cdot frac{1}{2} = a^2(a^2 + 1)^2 )2. ( 2a^2(a^2 + 1)^2 cdot left( -frac{1}{2a^2} right) = - (a^2 + 1)^2 )3. ( 2a^2(a^2 + 1)^2 cdot left( -frac{8(a^2 - 1)}{(a^2 + 1)^2} right) = -16a^2(a^2 - 1) )So, putting it all together:[a^2(a^2 + 1)^2 - (a^2 + 1)^2 - 16a^2(a^2 - 1) = 0]Factor out ( (a^2 + 1)^2 ):[(a^2 + 1)^2(a^2 - 1) - 16a^2(a^2 - 1) = 0]Factor out ( (a^2 - 1) ):[(a^2 - 1)left[ (a^2 + 1)^2 - 16a^2 right] = 0]So, either ( a^2 - 1 = 0 ) or ( (a^2 + 1)^2 - 16a^2 = 0 ).Case 1: ( a^2 - 1 = 0 )This gives ( a^2 = 1 ), so ( a = 1 ) (since ( a > 0 )).Case 2: ( (a^2 + 1)^2 - 16a^2 = 0 )Expand ( (a^2 + 1)^2 ):[a^4 + 2a^2 + 1 - 16a^2 = 0 implies a^4 - 14a^2 + 1 = 0]Let me set ( t = a^2 ), so the equation becomes:[t^2 - 14t + 1 = 0]Solve for ( t ):[t = frac{14 pm sqrt{196 - 4}}{2} = frac{14 pm sqrt{192}}{2} = frac{14 pm 8sqrt{3}}{2} = 7 pm 4sqrt{3}]Since ( t = a^2 > 0 ), both solutions are valid. So,[a^2 = 7 + 4sqrt{3} quad text{or} quad a^2 = 7 - 4sqrt{3}]Taking square roots:1. ( a = sqrt{7 + 4sqrt{3}} )2. ( a = sqrt{7 - 4sqrt{3}} )Simplify ( sqrt{7 + 4sqrt{3}} ):Notice that ( 7 + 4sqrt{3} = (2 + sqrt{3})^2 ), because:[(2 + sqrt{3})^2 = 4 + 4sqrt{3} + 3 = 7 + 4sqrt{3}]Similarly, ( 7 - 4sqrt{3} = (2 - sqrt{3})^2 ), because:[(2 - sqrt{3})^2 = 4 - 4sqrt{3} + 3 = 7 - 4sqrt{3}]So,1. ( a = 2 + sqrt{3} )2. ( a = 2 - sqrt{3} )But since ( a > 0 ), both are valid because ( 2 - sqrt{3} approx 2 - 1.732 = 0.267 > 0 ).So, the critical points are at ( a = 1 ), ( a = 2 + sqrt{3} ), and ( a = 2 - sqrt{3} ).Now, let's evaluate the original expression ( f(a) ) at these critical points to determine which gives the minimum value.First, at ( a = 1 ):Since ( ab = 1 ), ( b = 1 ).Compute each term:- ( frac{1}{2a} = frac{1}{2} )- ( frac{1}{2b} = frac{1}{2} )- ( frac{8}{a + b} = frac{8}{2} = 4 )So, total expression:[frac{1}{2} + frac{1}{2} + 4 = 5]Next, at ( a = 2 + sqrt{3} ):Since ( ab = 1 ), ( b = frac{1}{2 + sqrt{3}} ). Rationalize the denominator:[b = frac{1}{2 + sqrt{3}} cdot frac{2 - sqrt{3}}{2 - sqrt{3}} = frac{2 - sqrt{3}}{(2)^2 - (sqrt{3})^2} = frac{2 - sqrt{3}}{4 - 3} = 2 - sqrt{3}]So, ( b = 2 - sqrt{3} ).Compute each term:1. ( frac{1}{2a} = frac{1}{2(2 + sqrt{3})} )2. ( frac{1}{2b} = frac{1}{2(2 - sqrt{3})} )3. ( frac{8}{a + b} = frac{8}{(2 + sqrt{3}) + (2 - sqrt{3})} = frac{8}{4} = 2 )Simplify the first two terms:1. ( frac{1}{2(2 + sqrt{3})} = frac{1}{4 + 2sqrt{3}} ). Rationalize:[frac{1}{4 + 2sqrt{3}} cdot frac{4 - 2sqrt{3}}{4 - 2sqrt{3}} = frac{4 - 2sqrt{3}}{16 - 12} = frac{4 - 2sqrt{3}}{4} = 1 - frac{sqrt{3}}{2}]2. Similarly, ( frac{1}{2(2 - sqrt{3})} = frac{1}{4 - 2sqrt{3}} ). Rationalize:[frac{1}{4 - 2sqrt{3}} cdot frac{4 + 2sqrt{3}}{4 + 2sqrt{3}} = frac{4 + 2sqrt{3}}{16 - 12} = frac{4 + 2sqrt{3}}{4} = 1 + frac{sqrt{3}}{2}]So, adding the first two terms:[left(1 - frac{sqrt{3}}{2}right) + left(1 + frac{sqrt{3}}{2}right) = 2]Adding the third term:[2 + 2 = 4]So, the expression evaluates to 4 at ( a = 2 + sqrt{3} ).Similarly, at ( a = 2 - sqrt{3} ), since ( a ) and ( b ) are symmetric in the expression, the value will also be 4.Therefore, comparing the values at the critical points:- At ( a = 1 ): 5- At ( a = 2 + sqrt{3} ) and ( a = 2 - sqrt{3} ): 4Hence, the minimum value of the expression is 4.I think that's it. So, the minimum value is 4, achieved when ( a = 2 + sqrt{3} ) and ( b = 2 - sqrt{3} ), or vice versa.

💡Okay, so I have this problem from the Rhind Papyrus, which is one of the oldest math texts in the world. It's about dividing 120 loaves of bread into 5 portions that form an arithmetic sequence. Also, the sum of the three larger portions is exactly 7 times the sum of the two smaller portions. I need to find the smallest portion. The options are A: 4, B: 3, C: 2, D: 1.Alright, let's break this down. First, an arithmetic sequence means that each portion increases by a common difference. Let me denote the smallest portion as 'a' and the common difference as 'd'. So, the five portions would be: a, a + d, a + 2d, a + 3d, and a + 4d.Wait, actually, in the initial problem, the user mentioned using a-2d, a-d, a, a+d, a+2d. That's another way to represent an arithmetic sequence with five terms, centered around 'a'. Maybe that's a better approach because it keeps the middle term as 'a', making it symmetric. So, let's go with that: the portions are a - 2d, a - d, a, a + d, a + 2d.Now, the total sum of these portions is 120 loaves. So, let's add them up:(a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 5a.So, 5a = 120, which means a = 24. Okay, that's straightforward.Next, the problem states that the sum of the three larger portions is exactly 7 times the sum of the two smaller portions. Let's identify the three larger portions and the two smaller ones.The portions are: a - 2d, a - d, a, a + d, a + 2d.So, the two smaller portions are a - 2d and a - d, and the three larger portions are a, a + d, a + 2d.Wait, actually, hold on. The three larger portions should be the top three, which are a + 2d, a + d, and a. The two smaller portions are a - 2d and a - d.So, sum of the three larger portions: (a + 2d) + (a + d) + a = 3a + 3d.Sum of the two smaller portions: (a - 2d) + (a - d) = 2a - 3d.According to the problem, 3a + 3d = 7*(2a - 3d).Let me write that equation:3a + 3d = 7*(2a - 3d)Let me expand the right side:3a + 3d = 14a - 21dNow, let's bring all terms to one side:3a + 3d - 14a + 21d = 0Combine like terms:(3a - 14a) + (3d + 21d) = 0-11a + 24d = 0So, 24d = 11aWe already found that a = 24, so plug that in:24d = 11*24Divide both sides by 24:d = 11Okay, so the common difference d is 11.Now, let's find the smallest portion, which is a - 2d:a - 2d = 24 - 2*11 = 24 - 22 = 2So, the smallest portion is 2 loaves. That corresponds to option C.Wait, let me double-check my steps to make sure I didn't make a mistake.First, I set up the portions as a - 2d, a - d, a, a + d, a + 2d. That's correct for an arithmetic sequence with five terms.Sum is 5a = 120, so a = 24. That seems right.Sum of the three larger portions: (a + 2d) + (a + d) + a = 3a + 3d.Sum of the two smaller portions: (a - 2d) + (a - d) = 2a - 3d.Setting up the equation: 3a + 3d = 7*(2a - 3d). That's correct.Expanding: 3a + 3d = 14a - 21d.Bringing all terms to one side: -11a + 24d = 0.So, 24d = 11a. Since a = 24, d = (11*24)/24 = 11. That's correct.Smallest portion: a - 2d = 24 - 22 = 2. Yep, that's right.So, the answer is C: 2.

💡Okay, so I have this problem about parametric equations and polar coordinates. Let me try to figure it out step by step.First, part (1) asks for the general equation of line ( l ) and the Cartesian equation of curve ( C ). Starting with line ( l ). The parametric equations are given as:[begin{cases}x = a + frac{1}{2}t, y = 2a + frac{sqrt{3}}{2}t.end{cases}]I need to eliminate the parameter ( t ) to find the general equation. From the first equation, I can solve for ( t ):[x = a + frac{1}{2}t implies t = 2(x - a).]Now plug this into the second equation:[y = 2a + frac{sqrt{3}}{2} cdot 2(x - a) = 2a + sqrt{3}(x - a).]Simplify that:[y = 2a + sqrt{3}x - sqrt{3}a = sqrt{3}x + (2a - sqrt{3}a).]So, the general equation of line ( l ) is:[y = sqrt{3}x + a(2 - sqrt{3}).]Got that part.Now, for curve ( C ), the polar equation is ( rho = 2costheta ). I need to convert this to Cartesian coordinates. Recall that ( rho = sqrt{x^2 + y^2} ) and ( costheta = frac{x}{rho} ). So, substituting into the equation:[rho = 2costheta implies sqrt{x^2 + y^2} = 2 cdot frac{x}{sqrt{x^2 + y^2}}.]Multiply both sides by ( sqrt{x^2 + y^2} ):[x^2 + y^2 = 2x.]Rearrange to get the standard form:[x^2 - 2x + y^2 = 0 implies (x - 1)^2 + y^2 = 1.]So, curve ( C ) is a circle with center at ( (1, 0) ) and radius 1.Alright, part (1) seems done.Moving on to part (2). It says: If there is exactly one point on curve ( C ) that is at a distance of ( sqrt{3} - 1 ) from line ( l ), find the value of the real number ( a ).Hmm, okay. So, we have a circle ( C ) and a line ( l ). We need the distance from the line ( l ) to the circle ( C ) such that there's exactly one point on the circle at distance ( sqrt{3} - 1 ) from the line. Wait, actually, it's not the distance from the line to the circle, but the distance from points on the circle to the line. And there should be exactly one such point with that distance.I remember that the distance from a point ( (x_0, y_0) ) to the line ( Ax + By + C = 0 ) is given by:[text{Distance} = frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}}.]So, maybe I can use this formula for points on the circle and set the distance equal to ( sqrt{3} - 1 ).But since the circle is ( (x - 1)^2 + y^2 = 1 ), any point on the circle can be parameterized as ( (1 + costheta, sintheta) ).Alternatively, maybe it's easier to think about the distance from the center of the circle to the line, and then relate that to the radius.Let me recall: If the distance from the center to the line is ( d ), then the points on the circle will have distances from the line ranging from ( d - r ) to ( d + r ), where ( r ) is the radius.In our case, the radius ( r = 1 ). So, if there's exactly one point on the circle at distance ( sqrt{3} - 1 ) from the line, that suggests that this distance is either ( d - r ) or ( d + r ), and it's tangent to the circle.Wait, but the problem states that there is exactly one point on the curve ( C ) at that distance. So, that would mean that the line is tangent to a circle of radius ( sqrt{3} - 1 ) centered at the original circle's center? Hmm, maybe.Wait, no. Let me think again.The distance from the center of the circle to the line is ( d ). Then, the minimal distance from the line to any point on the circle is ( d - r ) and the maximal is ( d + r ). If there's exactly one point on the circle at a specific distance, say ( k ), then ( k ) must be equal to either ( d - r ) or ( d + r ). Because if ( k ) is between ( d - r ) and ( d + r ), there are two points on the circle at that distance from the line.But in our case, we have exactly one point, so ( k ) must be equal to ( d - r ) or ( d + r ). But since ( d ) is the distance from the center to the line, and ( r = 1 ), we have:Either ( k = d - 1 ) or ( k = d + 1 ).But the given distance is ( sqrt{3} - 1 ). So, let's set up the equation:Either ( d - 1 = sqrt{3} - 1 ) or ( d + 1 = sqrt{3} - 1 ).Solving these:1. ( d - 1 = sqrt{3} - 1 implies d = sqrt{3} ).2. ( d + 1 = sqrt{3} - 1 implies d = sqrt{3} - 2 ).But ( d ) is a distance, so it must be non-negative. ( sqrt{3} approx 1.732 ), so ( sqrt{3} - 2 approx -0.267 ), which is negative. So, this is not possible. Therefore, only the first case is valid: ( d = sqrt{3} ).So, the distance from the center of the circle ( (1, 0) ) to the line ( l ) must be ( sqrt{3} ).Now, let's compute the distance ( d ) from ( (1, 0) ) to the line ( l ).The general equation of line ( l ) is ( y = sqrt{3}x + a(2 - sqrt{3}) ). Let me write it in standard form ( Ax + By + C = 0 ):[sqrt{3}x - y + a(2 - sqrt{3}) = 0.]So, ( A = sqrt{3} ), ( B = -1 ), ( C = a(2 - sqrt{3}) ).The distance from ( (1, 0) ) to this line is:[d = frac{|sqrt{3}(1) + (-1)(0) + a(2 - sqrt{3})|}{sqrt{(sqrt{3})^2 + (-1)^2}} = frac{|sqrt{3} + a(2 - sqrt{3})|}{sqrt{3 + 1}} = frac{|sqrt{3} + a(2 - sqrt{3})|}{2}.]We know that this distance ( d = sqrt{3} ). So,[frac{|sqrt{3} + a(2 - sqrt{3})|}{2} = sqrt{3}.]Multiply both sides by 2:[|sqrt{3} + a(2 - sqrt{3})| = 2sqrt{3}.]This absolute value equation leads to two cases:1. ( sqrt{3} + a(2 - sqrt{3}) = 2sqrt{3} )2. ( sqrt{3} + a(2 - sqrt{3}) = -2sqrt{3} )Let's solve each case.**Case 1:**[sqrt{3} + a(2 - sqrt{3}) = 2sqrt{3}]Subtract ( sqrt{3} ) from both sides:[a(2 - sqrt{3}) = sqrt{3}]Solve for ( a ):[a = frac{sqrt{3}}{2 - sqrt{3}}.]To rationalize the denominator, multiply numerator and denominator by ( 2 + sqrt{3} ):[a = frac{sqrt{3}(2 + sqrt{3})}{(2 - sqrt{3})(2 + sqrt{3})} = frac{2sqrt{3} + 3}{4 - 3} = 2sqrt{3} + 3.]**Case 2:**[sqrt{3} + a(2 - sqrt{3}) = -2sqrt{3}]Subtract ( sqrt{3} ) from both sides:[a(2 - sqrt{3}) = -3sqrt{3}]Solve for ( a ):[a = frac{-3sqrt{3}}{2 - sqrt{3}}.]Again, rationalize the denominator:[a = frac{-3sqrt{3}(2 + sqrt{3})}{(2 - sqrt{3})(2 + sqrt{3})} = frac{-6sqrt{3} - 9}{1} = -6sqrt{3} - 9.]So, the possible values of ( a ) are ( 2sqrt{3} + 3 ) and ( -6sqrt{3} - 9 ).Wait, let me double-check my calculations. In Case 1, I had:[a = frac{sqrt{3}}{2 - sqrt{3}} times frac{2 + sqrt{3}}{2 + sqrt{3}} = frac{sqrt{3}(2 + sqrt{3})}{1} = 2sqrt{3} + 3.]Yes, that seems correct.In Case 2:[a = frac{-3sqrt{3}}{2 - sqrt{3}} times frac{2 + sqrt{3}}{2 + sqrt{3}} = frac{-3sqrt{3}(2 + sqrt{3})}{1} = -6sqrt{3} - 9.]That also looks correct.So, both solutions are valid. Therefore, the real number ( a ) can be either ( 2sqrt{3} + 3 ) or ( -6sqrt{3} - 9 ).I think that's it. Let me just recap:1. Converted parametric equations to general line equation.2. Converted polar equation to Cartesian, getting a circle.3. Understood that exactly one point on the circle at a certain distance implies the distance from the center to the line is equal to the given distance plus or minus the radius.4. Calculated the distance from the center to the line, set it equal to ( sqrt{3} ), solved for ( a ).5. Got two solutions, both valid.Yeah, that seems solid.**Final Answer**The value of the real number ( a ) is (boxed{-6sqrt{3} - 9}) or (boxed{2sqrt{3} + 3}).