💡Okay, so I have this problem about an isosceles trapezoid that's circumscribed about a circle. The longer base is 16, and one of the base angles is arcsin(0.8). I need to find the area of the trapezoid. Hmm, let me think about how to approach this.First, let me recall some properties of trapezoids and circles. An isosceles trapezoid has the legs equal in length and the base angles equal. Since it's circumscribed about a circle, it must be a tangential trapezoid, meaning it has an incircle. I remember that for a quadrilateral to be tangential, the sum of the lengths of the two opposite sides must be equal. So, in this case, the sum of the two bases should be equal to the sum of the two legs.Let me denote the trapezoid as ABCD, where AD is the longer base of length 16, and BC is the shorter base. The legs AB and CD are equal in length because it's isosceles. Let me denote the length of each leg as 'l'. Since it's tangential, AD + BC = AB + CD. Since AB = CD = l, this simplifies to AD + BC = 2l. So, 16 + BC = 2l. Therefore, BC = 2l - 16.Now, I need to relate this to the given angle. One of the base angles is arcsin(0.8). Let me denote this angle as θ, so θ = arcsin(0.8). That means sin(θ) = 0.8. Since it's a base angle, let's say angle at A is θ. In an isosceles trapezoid, the base angles are equal, so angle at D is also θ. The other two angles at B and C will be supplementary to θ, so they are 180° - θ.I can use trigonometry to find the height of the trapezoid. The height 'h' can be found using the sine of angle θ. If I drop a perpendicular from B to AD, let's call the foot of this perpendicular E. Similarly, drop a perpendicular from C to AD, call the foot F. Then, the segment AE is equal to DF, and the length of AE can be found using the cosine of θ, since in triangle ABE, cos(θ) = adjacent/hypotenuse = AE / AB. So, AE = AB * cos(θ).Similarly, the height h = AB * sin(θ). Since sin(θ) = 0.8, h = AB * 0.8.Now, the length of the shorter base BC can be found by subtracting twice the length AE from the longer base AD. So, BC = AD - 2*AE. Substituting AE = AB * cos(θ), we get BC = 16 - 2*AB*cos(θ).But earlier, we had BC = 2l - 16, where l is the length of the legs. Wait, but AB is a leg, so AB = l. So, substituting, BC = 2l - 16. Therefore, 16 - 2*l*cos(θ) = 2l - 16.Let me write that equation again:16 - 2*l*cos(θ) = 2l - 16Let me solve for l. First, bring all terms to one side:16 + 16 = 2l + 2*l*cos(θ)32 = 2l(1 + cos(θ))Divide both sides by 2:16 = l(1 + cos(θ))So, l = 16 / (1 + cos(θ))Now, I need to find cos(θ). Since sin(θ) = 0.8, I can use the Pythagorean identity: sin²(θ) + cos²(θ) = 1.So, cos²(θ) = 1 - sin²(θ) = 1 - 0.64 = 0.36. Therefore, cos(θ) = sqrt(0.36) = 0.6. Since θ is an acute angle in a trapezoid, cos(θ) is positive.So, cos(θ) = 0.6. Now, plug this back into the equation for l:l = 16 / (1 + 0.6) = 16 / 1.6 = 10.So, the length of each leg AB and CD is 10.Now, let's find the height h. h = AB * sin(θ) = 10 * 0.8 = 8.Now, let's find the shorter base BC. From earlier, BC = 16 - 2*l*cos(θ) = 16 - 2*10*0.6 = 16 - 12 = 4.So, the two bases are AD = 16 and BC = 4, and the height is 8.The area of a trapezoid is given by (1/2)*(sum of the bases)*height. So, area = (1/2)*(16 + 4)*8 = (1/2)*20*8 = 10*8 = 80.Wait, but let me double-check if I did everything correctly. Let me verify the steps.1. I used the property that in a tangential quadrilateral, the sum of the two opposite sides is equal. So, AD + BC = AB + CD. Since it's isosceles, AB = CD, so AD + BC = 2AB. That seems correct.2. I denoted AB as l, so BC = 2l - 16. Then, using trigonometry, I expressed BC in terms of l and θ, which is 16 - 2*l*cos(θ). Equating the two expressions for BC: 16 - 2*l*cos(θ) = 2l - 16. That led me to solve for l, which I found to be 10. Then, using sin(θ) = 0.8, I found the height h = 8. Then, BC = 4. Finally, the area is (1/2)*(16 + 4)*8 = 80.Yes, that seems consistent. I don't see any mistakes in the calculations. So, the area should be 80.**Final Answer**boxed{C}

💡Alright, so I have this problem: I need to figure out the relationship between the coefficients of two quadratic equations such that their roots form a geometric progression. Hmm, okay. Let me try to break this down step by step.First, let me recall what a quadratic equation looks like. It's generally in the form ( ax^2 + bx + c = 0 ), where ( a ), ( b ), and ( c ) are coefficients. The roots of this equation can be found using the quadratic formula, but maybe I don't need to go that far yet.Now, the problem mentions two quadratic equations. Let me denote them as:1. ( a_1x^2 + b_1x + c_1 = 0 )2. ( a_2x^2 + b_2x + c_2 = 0 )Each of these equations has two roots. Let's say the first equation has roots ( r_1 ) and ( r_2 ), and the second equation has roots ( r_3 ) and ( r_4 ). The problem states that all four roots ( r_1, r_2, r_3, r_4 ) form a geometric progression.Okay, so a geometric progression is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denoted as ( q ). So, if I let ( r_1 ) be the first term, then the sequence would be ( r_1, r_1q, r_1q^2, r_1q^3 ).But wait, the roots are split between two quadratic equations. That means the first equation has two roots, say ( r_1 ) and ( r_2 ), and the second equation has the other two roots, ( r_3 ) and ( r_4 ). So, the four roots in total are ( r_1, r_2, r_3, r_4 ), which are in geometric progression.So, let me assign them as follows:- ( r_1 = a )- ( r_2 = aq )- ( r_3 = aq^2 )- ( r_4 = aq^3 )Where ( a ) is the first term and ( q ) is the common ratio.Now, I need to relate this to the coefficients of the quadratic equations. For that, I can use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.For the first quadratic equation ( a_1x^2 + b_1x + c_1 = 0 ), the sum of the roots ( r_1 + r_2 = -frac{b_1}{a_1} ), and the product ( r_1r_2 = frac{c_1}{a_1} ).Similarly, for the second equation ( a_2x^2 + b_2x + c_2 = 0 ), the sum of the roots ( r_3 + r_4 = -frac{b_2}{a_2} ), and the product ( r_3r_4 = frac{c_2}{a_2} ).So, let me write down these relationships:1. ( r_1 + r_2 = a + aq = a(1 + q) = -frac{b_1}{a_1} )2. ( r_1r_2 = a cdot aq = a^2q = frac{c_1}{a_1} )3. ( r_3 + r_4 = aq^2 + aq^3 = aq^2(1 + q) = -frac{b_2}{a_2} )4. ( r_3r_4 = aq^2 cdot aq^3 = a^2q^5 = frac{c_2}{a_2} )Okay, so now I have four equations. Let me see how I can relate these to find the relationship between the coefficients ( a_1, b_1, c_1 ) and ( a_2, b_2, c_2 ).Looking at equations 1 and 3, both have the term ( a(1 + q) ) and ( aq^2(1 + q) ) respectively. Maybe I can express ( a(1 + q) ) from equation 1 and plug it into equation 3.From equation 1:( a(1 + q) = -frac{b_1}{a_1} )From equation 3:( aq^2(1 + q) = -frac{b_2}{a_2} )Notice that ( aq^2(1 + q) = q^2 cdot a(1 + q) ). So, substituting ( a(1 + q) ) from equation 1 into equation 3:( q^2 cdot left(-frac{b_1}{a_1}right) = -frac{b_2}{a_2} )Simplify this:( -frac{b_1 q^2}{a_1} = -frac{b_2}{a_2} )Multiply both sides by -1:( frac{b_1 q^2}{a_1} = frac{b_2}{a_2} )So, ( frac{b_1}{a_1} q^2 = frac{b_2}{a_2} )Let me write that as:( q^2 = frac{b_2 a_1}{b_1 a_2} )Okay, so that's one relationship involving ( q^2 ).Now, let's look at equations 2 and 4. Both involve the product of roots.From equation 2:( a^2 q = frac{c_1}{a_1} )From equation 4:( a^2 q^5 = frac{c_2}{a_2} )Let me express ( a^2 ) from equation 2:( a^2 = frac{c_1}{a_1 q} )Now, substitute this into equation 4:( left(frac{c_1}{a_1 q}right) q^5 = frac{c_2}{a_2} )Simplify:( frac{c_1}{a_1 q} cdot q^5 = frac{c_2}{a_2} )( frac{c_1 q^4}{a_1} = frac{c_2}{a_2} )So, ( frac{c_1}{a_1} q^4 = frac{c_2}{a_2} )From this, we can write:( q^4 = frac{c_2 a_1}{c_1 a_2} )Now, earlier we had ( q^2 = frac{b_2 a_1}{b_1 a_2} ). If I square both sides of this equation, I get:( q^4 = left(frac{b_2 a_1}{b_1 a_2}right)^2 )But from the product of roots, we also have ( q^4 = frac{c_2 a_1}{c_1 a_2} ). Therefore, these two expressions for ( q^4 ) must be equal:( left(frac{b_2 a_1}{b_1 a_2}right)^2 = frac{c_2 a_1}{c_1 a_2} )Let me write that out:( left(frac{b_2 a_1}{b_1 a_2}right)^2 = frac{c_2 a_1}{c_1 a_2} )Simplify the left side:( frac{b_2^2 a_1^2}{b_1^2 a_2^2} = frac{c_2 a_1}{c_1 a_2} )Now, let's cross-multiply to eliminate denominators:( b_2^2 a_1^2 cdot c_1 a_2 = c_2 a_1 cdot b_1^2 a_2^2 )Simplify both sides:Left side: ( b_2^2 a_1^2 c_1 a_2 )Right side: ( c_2 a_1 b_1^2 a_2^2 )Divide both sides by ( a_1 a_2 ):Left side: ( b_2^2 a_1 c_1 )Right side: ( c_2 b_1^2 a_2 )So, we have:( b_2^2 a_1 c_1 = c_2 b_1^2 a_2 )Let me rearrange this equation to express the relationship between the coefficients:( frac{b_2^2}{b_1^2} = frac{c_2 a_2}{c_1 a_1} )Alternatively, taking square roots on both sides (though I think it's better to keep it squared for clarity):( left(frac{b_2}{b_1}right)^2 = frac{c_2 a_2}{c_1 a_1} )Wait, hold on. Let me double-check the cross-multiplication step because I might have messed up the exponents.Original equation after cross-multiplying:( b_2^2 a_1^2 c_1 a_2 = c_2 a_1 b_1^2 a_2^2 )Yes, that's correct. Then dividing both sides by ( a_1 a_2 ):Left: ( b_2^2 a_1 c_1 )Right: ( c_2 b_1^2 a_2 )Yes, so:( b_2^2 a_1 c_1 = c_2 b_1^2 a_2 )Therefore, rearranged:( frac{b_2^2}{b_1^2} = frac{c_2 a_2}{c_1 a_1} )Wait, but in the previous step, I had:( left(frac{b_2 a_1}{b_1 a_2}right)^2 = frac{c_2 a_1}{c_1 a_2} )Which simplifies to:( frac{b_2^2 a_1^2}{b_1^2 a_2^2} = frac{c_2 a_1}{c_1 a_2} )Cross-multiplying:( b_2^2 a_1^2 c_1 a_2 = c_2 a_1 b_1^2 a_2^2 )Divide both sides by ( a_1 a_2 ):( b_2^2 a_1 c_1 = c_2 b_1^2 a_2 )So, yes, that's correct.Therefore, the relationship is:( frac{b_2^2}{b_1^2} = frac{c_2 a_2}{c_1 a_1} )Alternatively, this can be written as:( left(frac{b_2}{b_1}right)^2 = frac{c_2 a_2}{c_1 a_1} )Or, to make it more symmetric:( frac{b_1^2}{b_2^2} = frac{c_1 a_1}{c_2 a_2} )Either way, this is the relationship that must hold between the coefficients of the two quadratic equations for their roots to form a geometric progression.Let me just recap to make sure I didn't miss anything. I started by assigning the roots as a geometric sequence, used Vieta's formulas to relate the sums and products of roots to the coefficients, and then derived the relationship by equating expressions for ( q^4 ) from both the sum and product of roots. It seems consistent.I think that's the required relationship. So, the key is that the square of the ratio of the linear coefficients equals the ratio of the product of the constant term and the leading coefficient of the second equation to the same for the first equation.**Final Answer**The required relationship is boxed{left( frac{b_1}{b_2} right)^2 = frac{c_1 a_1}{c_2 a_2}}.

💡First, I need to determine the ratio of Suresh's and Ramesh's investments. Suresh invested ₹24,000 and Ramesh invested ₹40,000.To simplify the ratio, I'll divide both amounts by their greatest common divisor, which is ₹8,000. This gives me a simplified ratio of 3:5.Next, I'll calculate the total number of parts in the ratio, which is 3 + 5 = 8 parts.Ramesh's share corresponds to 5 parts out of the total 8 parts. To find Ramesh's share of the profit, I'll multiply the total profit by the fraction of the ratio that represents Ramesh's investment.Finally, I'll calculate Ramesh's share by multiplying ₹19,000 by 5/8, resulting in ₹11,875.

💡First, I need to determine the number of frisbees sold at each price point. Let ( x ) represent the number of 3 frisbees sold, and ( y ) represent the number of 4 frisbees sold.I know that the total number of frisbees sold is 60, so I can write the equation:[x + y = 60]The problem states that the fewest number of 4 frisbees sold is 24, which means ( y geq 24 ).To find the total receipts, I need to calculate the revenue from both 3 and 4 frisbees. If the minimum number of 4 frisbees sold is 24, then:[y = 24]Substituting this into the first equation to find ( x ):[x + 24 = 60 x = 60 - 24 x = 36]Now, I can calculate the total receipts:[text{Total receipts} = (36 times 3) + (24 times 4) text{Total receipts} = 108 + 96 text{Total receipts} = 204]

💡Okay, so I have this problem where I need to find the smallest positive integer greater than 1 that leaves a remainder of 1 when divided by each of the numbers 2, 3, 4, 5, 6, 7, 8, and 9. Hmm, that sounds a bit tricky, but let me try to break it down step by step.First, I need to understand what it means for a number to leave a remainder of 1 when divided by another number. If a number, let's call it N, leaves a remainder of 1 when divided by, say, 2, that means when I divide N by 2, I get some quotient and a remainder of 1. So, mathematically, that can be written as N = 2k + 1, where k is some integer. Similarly, for 3, it would be N = 3m + 1, and so on for all the numbers up to 9.Wait a minute, so if N leaves a remainder of 1 when divided by all these numbers, then N - 1 must be divisible by all of them without any remainder. That makes sense because if N = divisor * quotient + 1, then N - 1 = divisor * quotient, which means N - 1 is a multiple of that divisor.So, if N - 1 is divisible by 2, 3, 4, 5, 6, 7, 8, and 9, then N - 1 must be a common multiple of all these numbers. To find the smallest such N, I need to find the least common multiple (LCM) of these numbers and then add 1 to it.But how do I find the LCM of multiple numbers? I remember that the LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. To find the LCM, I can break down each number into its prime factors and then take the highest power of each prime that appears in any of the factorizations.Let me list out the numbers and their prime factorizations:- 2 is already a prime number: 2- 3 is also a prime number: 3- 4 can be factored into 2 * 2, which is 2²- 5 is a prime number: 5- 6 can be factored into 2 * 3- 7 is a prime number: 7- 8 can be factored into 2 * 2 * 2, which is 2³- 9 can be factored into 3 * 3, which is 3²Now, to find the LCM, I need to take the highest power of each prime number that appears in these factorizations:- For 2, the highest power is 2³ (from 8)- For 3, the highest power is 3² (from 9)- For 5, it's just 5¹- For 7, it's just 7¹So, the LCM would be 2³ * 3² * 5 * 7.Let me calculate that:First, 2³ is 8.Then, 3² is 9.Now, multiplying these together: 8 * 9 = 72.Next, multiply by 5: 72 * 5 = 360.Finally, multiply by 7: 360 * 7 = 2520.So, the LCM of 2, 3, 4, 5, 6, 7, 8, and 9 is 2520.That means N - 1 = 2520, so N = 2520 + 1 = 2521.Let me double-check to make sure this works. If I divide 2521 by each of these numbers, does it leave a remainder of 1?- 2521 ÷ 2 = 1260 with a remainder of 1- 2521 ÷ 3 = 840 with a remainder of 1- 2521 ÷ 4 = 630 with a remainder of 1- 2521 ÷ 5 = 504 with a remainder of 1- 2521 ÷ 6 = 420 with a remainder of 1- 2521 ÷ 7 = 360 with a remainder of 1- 2521 ÷ 8 = 315 with a remainder of 1- 2521 ÷ 9 = 280 with a remainder of 1Yes, it works for all of them. So, 2521 is indeed the smallest number greater than 1 that leaves a remainder of 1 when divided by each of these numbers.I think I've got it right. The key was realizing that N - 1 needs to be a common multiple of all those divisors, and finding the least common multiple was the way to go. Breaking down each number into its prime factors helped me calculate the LCM accurately. It's always good to double-check by actually performing the divisions to ensure the remainder is indeed 1 each time. This problem was a good exercise in understanding least common multiples and how they apply to real-world problems.

💡Okay, so I have this problem here: I need to find the smallest positive integer ( n ) such that there exist positive real numbers ( c ) and ( d ) where ( (c + di)^{2n} = (c - di)^{2n} ). Then, I have to compute the ratio ( frac{d}{c} ).Hmm, let me start by understanding what this equation means. ( c + di ) and ( c - di ) are complex conjugates of each other. So, when I raise them to some power, I want their results to be equal. That seems interesting because usually, complex conjugates raised to powers can give different results unless certain conditions are met.Let me write down the equation again:[(c + di)^{2n} = (c - di)^{2n}]Since ( c ) and ( d ) are positive real numbers, ( c + di ) is a complex number in the first quadrant of the complex plane, and ( c - di ) is its reflection across the real axis.I remember that for complex numbers, if two complex numbers are equal, their magnitudes must be equal, and their arguments must differ by an integer multiple of ( 2pi ). So, maybe I can express ( c + di ) and ( c - di ) in polar form and then analyze their powers.Let me denote ( z = c + di ). Then, ( overline{z} = c - di ), which is the complex conjugate of ( z ). So, the equation becomes:[z^{2n} = (overline{z})^{2n}]Which can be rewritten as:[z^{2n} = overline{z}^{2n}]Taking the modulus on both sides, since modulus is multiplicative, we have:[|z|^{2n} = |overline{z}|^{2n}]But ( |z| = |overline{z}| ), so this equation is always true. That doesn't give me any new information. So, I need to look at the arguments.Let me write ( z ) in polar form. Let ( z = r e^{itheta} ), where ( r = sqrt{c^2 + d^2} ) is the modulus, and ( theta = arctanleft(frac{d}{c}right) ) is the argument. Then, ( overline{z} = r e^{-itheta} ).So, substituting into the equation:[(r e^{itheta})^{2n} = (r e^{-itheta})^{2n}]Simplifying both sides:[r^{2n} e^{i 2n theta} = r^{2n} e^{-i 2n theta}]Since ( r ) is positive, ( r^{2n} ) is non-zero, so we can divide both sides by ( r^{2n} ):[e^{i 2n theta} = e^{-i 2n theta}]This implies that:[e^{i 4n theta} = 1]Because if ( e^{i alpha} = e^{-i alpha} ), then ( e^{i 2alpha} = 1 ). So, ( 4n theta ) must be an integer multiple of ( 2pi ):[4n theta = 2pi k]Where ( k ) is an integer. Simplifying:[2n theta = pi k]So,[theta = frac{pi k}{2n}]But ( theta ) is the argument of ( z = c + di ), which is ( arctanleft(frac{d}{c}right) ). Since ( c ) and ( d ) are positive, ( theta ) is in the first quadrant, so ( 0 < theta < frac{pi}{2} ).Therefore, ( frac{pi k}{2n} ) must be in ( (0, frac{pi}{2}) ). So,[0 < frac{pi k}{2n} < frac{pi}{2}]Dividing through by ( pi ):[0 < frac{k}{2n} < frac{1}{2}]Multiplying all terms by ( 2n ):[0 < k < n]Since ( k ) is an integer, the possible values of ( k ) are ( 1, 2, ldots, n-1 ).Now, we need to find the smallest positive integer ( n ) such that there exists an integer ( k ) with ( 1 leq k < n ) and ( theta = frac{pi k}{2n} ).But ( theta = arctanleft(frac{d}{c}right) ), so:[arctanleft(frac{d}{c}right) = frac{pi k}{2n}]Taking the tangent of both sides:[frac{d}{c} = tanleft(frac{pi k}{2n}right)]So, the ratio ( frac{d}{c} ) is equal to the tangent of ( frac{pi k}{2n} ).Now, our goal is to find the smallest ( n ) such that this equation holds for some positive real numbers ( c ) and ( d ). Since ( c ) and ( d ) are positive, ( frac{d}{c} ) is positive, so ( tanleft(frac{pi k}{2n}right) ) must be positive, which it is since ( 0 < frac{pi k}{2n} < frac{pi}{2} ).So, the key is to find the smallest ( n ) such that ( frac{pi k}{2n} ) is an angle whose tangent is a positive real number, which is always true for the given range. But perhaps more importantly, we need ( theta ) to be such that ( z^{2n} = overline{z}^{2n} ).Wait, but we already derived that condition, so maybe the key is to find the minimal ( n ) such that ( 2n theta ) is an integer multiple of ( pi ). Because from earlier, ( 2n theta = pi k ), so ( theta = frac{pi k}{2n} ).But ( theta ) is ( arctanleft(frac{d}{c}right) ), so ( frac{d}{c} = tanleft(frac{pi k}{2n}right) ).So, for each ( n ), we can choose ( k ) such that ( 1 leq k < n ), and then ( frac{d}{c} ) is determined.But the problem states that ( n ) is the smallest positive integer for which such ( c ) and ( d ) exist. So, I need to find the minimal ( n ) such that there exists an integer ( k ) with ( 1 leq k < n ) and ( tanleft(frac{pi k}{2n}right) ) is a positive real number, which it always is. But perhaps more specifically, we need to ensure that ( z^{2n} ) is equal to ( overline{z}^{2n} ), which as we saw, requires that ( 4n theta ) is a multiple of ( 2pi ), so ( 2n theta ) is a multiple of ( pi ).Wait, let me double-check that. From earlier:We had ( e^{i 4n theta} = 1 ), which implies ( 4n theta = 2pi k ), so ( 2n theta = pi k ). So, ( theta = frac{pi k}{2n} ).Therefore, ( theta ) must be a rational multiple of ( pi ). So, the angle ( theta ) must be such that it's a rational multiple of ( pi ), which is always true because ( k ) and ( n ) are integers.But perhaps the key is that ( frac{pi k}{2n} ) must be such that ( tanleft(frac{pi k}{2n}right) ) is a real number, which it is, but maybe we need ( frac{pi k}{2n} ) to be an angle that can be expressed as ( arctanleft(frac{d}{c}right) ) for some positive ( c ) and ( d ). But since ( c ) and ( d ) are positive, ( theta ) is in the first quadrant, so ( frac{pi k}{2n} ) must be in ( (0, frac{pi}{2}) ).So, the question is, what is the minimal ( n ) such that there exists an integer ( k ) with ( 1 leq k < n ) and ( frac{pi k}{2n} ) is in ( (0, frac{pi}{2}) ). Well, for any ( n geq 1 ), we can choose ( k = 1 ), which gives ( theta = frac{pi}{2n} ), which is in ( (0, frac{pi}{2}) ) for ( n geq 1 ).Wait, but that seems too easy. Because for ( n = 1 ), ( k = 1 ), ( theta = frac{pi}{2} ), but ( theta ) must be less than ( frac{pi}{2} ). So, ( n = 1 ) would give ( theta = frac{pi}{2} ), which is not allowed because ( theta ) must be strictly less than ( frac{pi}{2} ). So, ( n = 1 ) is invalid.Wait, let me check ( n = 1 ). If ( n = 1 ), then ( k ) must be less than ( n ), so ( k = 0 ), but ( k ) must be at least 1. So, actually, for ( n = 1 ), there is no such ( k ) because ( k ) must satisfy ( 1 leq k < 1 ), which is impossible. Therefore, ( n = 1 ) is invalid.So, moving on to ( n = 2 ). For ( n = 2 ), ( k ) can be 1. So, ( theta = frac{pi times 1}{2 times 2} = frac{pi}{4} ). So, ( frac{d}{c} = tanleft(frac{pi}{4}right) = 1 ). So, ( frac{d}{c} = 1 ). Therefore, ( c = d ).Wait, but let me verify this with ( n = 2 ). Let me compute ( (c + di)^4 ) and ( (c - di)^4 ) and see if they are equal when ( c = d ).Let ( c = d ). Then, ( z = c + ci = c(1 + i) ). So, ( z^4 = [c(1 + i)]^4 = c^4 (1 + i)^4 ).Compute ( (1 + i)^4 ):( (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i ).Then, ( (1 + i)^4 = (2i)^2 = -4 ).Similarly, ( (c - di)^4 = [c(1 - i)]^4 = c^4 (1 - i)^4 ).Compute ( (1 - i)^4 ):( (1 - i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i ).Then, ( (1 - i)^4 = (-2i)^2 = -4 ).So, both ( (c + di)^4 ) and ( (c - di)^4 ) equal ( -4c^4 ), which are equal. So, for ( n = 2 ), it works when ( c = d ).But wait, is ( n = 2 ) the minimal ( n )? Let's check ( n = 1 ) again. For ( n = 1 ), the equation becomes ( (c + di)^2 = (c - di)^2 ).Compute both sides:( (c + di)^2 = c^2 + 2cdi - d^2 ).( (c - di)^2 = c^2 - 2cdi - d^2 ).Setting them equal:( c^2 + 2cdi - d^2 = c^2 - 2cdi - d^2 ).Subtracting ( c^2 - d^2 ) from both sides:( 4cdi = 0 ).But ( c ) and ( d ) are positive, so ( 4cdi ) cannot be zero. Therefore, no solution exists for ( n = 1 ).So, ( n = 2 ) is indeed the minimal ( n ). Therefore, the ratio ( frac{d}{c} = 1 ).Wait, but let me think again. Is there a smaller ( n ) than 2? Well, ( n ) has to be a positive integer, so the next possible is ( n = 1 ), which doesn't work, so ( n = 2 ) is the smallest.But just to be thorough, let me consider ( n = 3 ). For ( n = 3 ), ( k ) can be 1 or 2.If ( k = 1 ), ( theta = frac{pi}{6} ), so ( frac{d}{c} = tanleft(frac{pi}{6}right) = frac{1}{sqrt{3}} ).If ( k = 2 ), ( theta = frac{pi}{3} ), so ( frac{d}{c} = tanleft(frac{pi}{3}right) = sqrt{3} ).But since ( n = 2 ) already works, ( n = 3 ) is not minimal. So, the minimal ( n ) is 2, and the ratio ( frac{d}{c} = 1 ).Therefore, the answer is ( boxed{1} ).