💡Okay, so I have this problem here, and I need to figure out the sum of all integers m that satisfy two conditions. The first condition is that the fractional equation (mx - 1)/(x - 2) = 2 + 3/(2 - x) has an integer solution. The second condition is that the graph of the quadratic function y = (m - 2)x² + 2x + 1 intersects the x-axis. Hmm, that sounds a bit complicated, but let me break it down step by step.First, I'll tackle the fractional equation. The equation is (mx - 1)/(x - 2) = 2 + 3/(2 - x). I notice that the denominators are x - 2 and 2 - x, which are negatives of each other. Maybe I can rewrite the equation to make it easier to handle. Let me see, 3/(2 - x) is the same as -3/(x - 2). So, substituting that in, the equation becomes (mx - 1)/(x - 2) = 2 - 3/(x - 2).Now, both sides have the same denominator, so maybe I can combine them. Let me rewrite the right side with a common denominator. That would be [2(x - 2) - 3]/(x - 2). Let me compute that numerator: 2(x - 2) is 2x - 4, and subtracting 3 gives 2x - 7. So, the equation simplifies to (mx - 1)/(x - 2) = (2x - 7)/(x - 2).Since the denominators are the same and assuming x ≠ 2 (because that would make the denominator zero), I can set the numerators equal to each other: mx - 1 = 2x - 7.Now, solving for x: mx - 1 = 2x - 7. Let me bring all terms to one side: mx - 2x = -7 + 1, which simplifies to (m - 2)x = -6.So, x = -6/(m - 2). But the problem states that the equation has an integer solution. That means x must be an integer, so -6/(m - 2) must be an integer. Therefore, m - 2 must be a divisor of -6.Let me list all the integer divisors of -6. The divisors are ±1, ±2, ±3, ±6. So, m - 2 can be 1, -1, 2, -2, 3, -3, 6, -6. Therefore, m can be 2 + 1 = 3, 2 - 1 = 1, 2 + 2 = 4, 2 - 2 = 0, 2 + 3 = 5, 2 - 3 = -1, 2 + 6 = 8, 2 - 6 = -4.Wait, but hold on. The original equation has a denominator x - 2, so x cannot be 2. So, I need to make sure that x = -6/(m - 2) ≠ 2. Let me check each value of m to ensure that x ≠ 2.For m = 3: x = -6/(3 - 2) = -6/1 = -6 ≠ 2. Okay, that's fine.For m = 1: x = -6/(1 - 2) = -6/(-1) = 6 ≠ 2. Good.For m = 4: x = -6/(4 - 2) = -6/2 = -3 ≠ 2. Okay.For m = 0: x = -6/(0 - 2) = -6/(-2) = 3 ≠ 2. Good.For m = 5: x = -6/(5 - 2) = -6/3 = -2 ≠ 2. Okay.For m = -1: x = -6/(-1 - 2) = -6/(-3) = 2. Oh, wait, x = 2 here, which is not allowed because it makes the denominator zero. So, m = -1 is invalid.For m = 8: x = -6/(8 - 2) = -6/6 = -1 ≠ 2. Good.For m = -4: x = -6/(-4 - 2) = -6/(-6) = 1 ≠ 2. Okay.So, the valid m values are 3, 1, 4, 0, 5, 8, -4. Wait, but earlier I thought m = -1 was invalid because x = 2, so I should exclude m = -1. So, m can be 3, 1, 4, 0, 5, 8, -4.Wait, but let me double-check m = 5: x = -2, which is fine. m = 8: x = -1, also fine. m = -4: x = 1, which is okay. m = 0: x = 3, okay. m = 1: x = 6, okay. m = 3: x = -6, okay. m = 4: x = -3, okay. So, all these m values except m = -1 are valid.So, the possible m values are 3, 1, 4, 0, 5, 8, -4.Now, moving on to the second condition: the graph of the quadratic function y = (m - 2)x² + 2x + 1 intersects the x-axis. For a quadratic function to intersect the x-axis, its discriminant must be non-negative. The discriminant D is given by D = b² - 4ac.In this case, a = (m - 2), b = 2, c = 1. So, D = (2)² - 4*(m - 2)*(1) = 4 - 4(m - 2).Simplify that: 4 - 4m + 8 = 12 - 4m.For the quadratic to intersect the x-axis, D ≥ 0: 12 - 4m ≥ 0.Solving for m: 12 ≥ 4m ⇒ m ≤ 3.Also, since it's a quadratic function, the coefficient of x² must not be zero, so m - 2 ≠ 0 ⇒ m ≠ 2.So, combining these, m must be less than or equal to 3 and not equal to 2.Now, from the first condition, the possible m values are 3, 1, 4, 0, 5, 8, -4.But from the second condition, m must be ≤ 3 and m ≠ 2.So, let's see which of the m values from the first condition satisfy m ≤ 3 and m ≠ 2.Looking at the list: 3, 1, 4, 0, 5, 8, -4.So, m = 3: 3 ≤ 3, okay.m = 1: 1 ≤ 3, okay.m = 4: 4 > 3, so exclude.m = 0: 0 ≤ 3, okay.m = 5: 5 > 3, exclude.m = 8: 8 > 3, exclude.m = -4: -4 ≤ 3, okay.So, the valid m values are 3, 1, 0, -4.Wait, but m = 3: let's check if m = 3 is valid. From the first condition, m = 3 gives x = -6/(3 - 2) = -6, which is fine. From the second condition, m = 3: D = 12 - 4*3 = 12 - 12 = 0, which is okay because D ≥ 0.Similarly, m = 1: D = 12 - 4*1 = 8 ≥ 0.m = 0: D = 12 - 4*0 = 12 ≥ 0.m = -4: D = 12 - 4*(-4) = 12 + 16 = 28 ≥ 0.So, all these m values satisfy both conditions.Wait, but earlier I thought m = 5 was excluded because m ≤ 3, but m = 5 is indeed greater than 3, so it's excluded. Similarly, m = 4, 8 are excluded.So, the valid m values are 3, 1, 0, -4.Now, the problem asks for the sum of all integers m that satisfy both conditions. So, let's add them up: 3 + 1 + 0 + (-4) = 3 + 1 = 4, 4 + 0 = 4, 4 + (-4) = 0.So, the sum is 0.Wait, but let me double-check if I missed any m values.From the first condition, the possible m values are 3, 1, 4, 0, 5, 8, -4.From the second condition, m must be ≤ 3 and m ≠ 2.So, m = 3, 1, 0, -4 are valid.Yes, that's correct. So, the sum is 0.I think that's the answer.

💡Okay, so I have this problem where I need to find the values of a, b, and c for the equation x² + y² + 2ax - by + c = 0. The equation is supposed to represent a circle with center at C(2, 3) and radius 3. Hmm, I remember that the general equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Alright, so let me write down the given information. The center is (2, 3), so h = 2 and k = 3. The radius is 3, so r = 3. Plugging these into the standard circle equation, I get:(x - 2)² + (y - 3)² = 3²Which simplifies to:(x - 2)² + (y - 3)² = 9Now, I need to expand this equation to match the form given in the problem, which is x² + y² + 2ax - by + c = 0. Let me expand the left side:First, expand (x - 2)²:(x - 2)² = x² - 4x + 4Next, expand (y - 3)²:(y - 3)² = y² - 6y + 9Now, add these two results together:x² - 4x + 4 + y² - 6y + 9 = 9Combine like terms:x² + y² - 4x - 6y + 4 + 9 = 9Simplify the constants:x² + y² - 4x - 6y + 13 = 9Now, subtract 9 from both sides to set the equation to zero:x² + y² - 4x - 6y + 4 = 0So, comparing this with the given form x² + y² + 2ax - by + c = 0, I can set up the following equations by matching coefficients:1. The coefficient of x is -4, which should be equal to 2a. So, 2a = -4. Solving for a, I divide both sides by 2: a = -2.2. The coefficient of y is -6, which should be equal to -b. So, -b = -6. Solving for b, I multiply both sides by -1: b = 6.3. The constant term is 4, which should be equal to c. So, c = 4.Let me double-check my work to make sure I didn't make any mistakes. Starting from the standard form:(x - 2)² + (y - 3)² = 9Expanding:x² - 4x + 4 + y² - 6y + 9 = 9Combine like terms:x² + y² - 4x - 6y + 13 = 9Subtract 9:x² + y² - 4x - 6y + 4 = 0Yes, that looks correct. So, comparing with x² + y² + 2ax - by + c = 0:- Coefficient of x: -4 = 2a ⇒ a = -2- Coefficient of y: -6 = -b ⇒ b = 6- Constant term: 4 = c ⇒ c = 4Everything checks out. I think I've got the right values for a, b, and c.

💡First, I need to determine the total cost of purchasing 15 apps at 4 each without any discount. This can be calculated by multiplying the number of apps by the cost per app.Next, I will calculate the discount amount by applying the 15% discount rate to the total cost of the apps. This involves converting the percentage to a decimal and multiplying it by the total cost.After finding the discount amount, I will subtract it from the total cost to determine the final amount Lidia needs to pay for the apps.Finally, I will subtract the total cost with the discount from the amount of money Lidia has to find out how much money she will have left after purchasing the apps.

💡Alright, so I have this problem where Thaddeus is given a 2013x2013 array of integers, each between 1 and 2013 inclusive. He can perform two operations: subtract 1 from every entry in a row or add 1 to every entry in a column. His goal is to make every number in the array divisible by 2013. I need to figure out how many such arrays are possible for him to achieve this.First, let me try to understand the operations. If I subtract 1 from an entire row, it's like decreasing each element in that row by 1. Similarly, adding 1 to a column increases each element in that column by 1. So, these operations can adjust the values in the array, but they affect entire rows or columns at a time.Since the goal is to make every number divisible by 2013, I need to think about this in terms of modular arithmetic. Specifically, modulo 2013. If a number is divisible by 2013, it means that number is congruent to 0 modulo 2013. So, for each entry in the array, I need to adjust it such that it becomes 0 modulo 2013.Let me denote the original array as ( A = [a_{i,j}] ) where ( 1 leq i, j leq 2013 ). Each ( a_{i,j} ) is between 1 and 2013. Thaddeus can perform row operations and column operations. Let's think about how these operations affect the entries modulo 2013.Suppose he subtracts 1 from row ( i ). Then each ( a_{i,j} ) becomes ( a_{i,j} - 1 ). Similarly, if he adds 1 to column ( j ), each ( a_{i,j} ) becomes ( a_{i,j} + 1 ). Since we're working modulo 2013, subtracting 1 is equivalent to adding 2012 modulo 2013, and adding 1 is just adding 1 modulo 2013.Let me formalize this. Let ( r_i ) be the number of times we subtract 1 from row ( i ), and ( c_j ) be the number of times we add 1 to column ( j ). Then, the final value of ( a_{i,j} ) after all operations will be:[ a_{i,j} - r_i + c_j equiv 0 pmod{2013} ]This simplifies to:[ a_{i,j} equiv r_i - c_j pmod{2013} ]So, for each entry ( a_{i,j} ), the difference between the number of row subtractions and column additions must be congruent to ( a_{i,j} ) modulo 2013.Now, let's think about the system of congruences this creates. For each ( i ) and ( j ), we have:[ r_i - c_j equiv a_{i,j} pmod{2013} ]This is a system of equations in the variables ( r_i ) and ( c_j ). I need to determine whether such a system has a solution for a given array ( A ), and if so, how many such arrays ( A ) allow for a solution.To analyze this, let's fix some variables. Let's fix ( r_1 ) and ( c_1 ). Then, from the first row and first column, we can express all other ( r_i ) and ( c_j ) in terms of ( r_1 ) and ( c_1 ).From the first row (( i = 1 )):[ r_1 - c_j equiv a_{1,j} pmod{2013} ]So,[ c_j equiv r_1 - a_{1,j} pmod{2013} ]Similarly, from the first column (( j = 1 )):[ r_i - c_1 equiv a_{i,1} pmod{2013} ]So,[ r_i equiv a_{i,1} + c_1 pmod{2013} ]Now, let's substitute these expressions into the general equation for ( a_{i,j} ):[ r_i - c_j equiv a_{i,j} pmod{2013} ]Substituting ( r_i ) and ( c_j ):[ (a_{i,1} + c_1) - (r_1 - a_{1,j}) equiv a_{i,j} pmod{2013} ]Simplify:[ a_{i,1} + c_1 - r_1 + a_{1,j} equiv a_{i,j} pmod{2013} ]Rearrange:[ a_{i,j} equiv a_{i,1} + a_{1,j} + (c_1 - r_1) pmod{2013} ]This equation must hold for all ( i, j ). Let me denote ( k = c_1 - r_1 ). Then,[ a_{i,j} equiv a_{i,1} + a_{1,j} + k pmod{2013} ]This suggests that the value of ( a_{i,j} ) is determined by the values in the first row and first column, plus a constant ( k ). Now, for this to hold for all ( i, j ), the array ( A ) must satisfy the condition that:[ a_{i,j} - a_{i,1} - a_{1,j} equiv k pmod{2013} ]for some constant ( k ). This means that the differences ( a_{i,j} - a_{i,1} - a_{1,j} ) must be the same for all ( i, j ).Therefore, the array ( A ) must satisfy the condition that for all ( i, j ), ( a_{i,j} equiv a_{i,1} + a_{1,j} + k pmod{2013} ) for some constant ( k ).This is a key insight. It tells me that the array is determined by the first row, the first column, and a constant ( k ). Therefore, the number of such arrays is determined by the number of choices for the first row, the first column, and the constant ( k ).However, I need to ensure that the array ( A ) is consistent. That is, the values in the first row and first column must be chosen such that the rest of the array satisfies the condition above.Let me think about how to count the number of such arrays. First, the first row has 2013 entries, each between 1 and 2013. Similarly, the first column has 2013 entries, but note that the first entry ( a_{1,1} ) is shared between the first row and the first column. So, we have to be careful not to double-count ( a_{1,1} ).Let me denote the first row as ( a_{1,1}, a_{1,2}, ldots, a_{1,2013} ) and the first column as ( a_{1,1}, a_{2,1}, ldots, a_{2013,1} ).Given that ( a_{i,j} equiv a_{i,1} + a_{1,j} + k pmod{2013} ), we can see that once we fix the first row and the first column, the rest of the array is determined by the constant ( k ).But wait, ( k ) is also determined by the first row and first column. Specifically, from ( a_{1,1} ), we have:[ a_{1,1} equiv a_{1,1} + a_{1,1} + k pmod{2013} ]Simplifying:[ a_{1,1} equiv 2a_{1,1} + k pmod{2013} ]Which implies:[ k equiv -a_{1,1} pmod{2013} ]So, ( k ) is uniquely determined by ( a_{1,1} ). Therefore, once we fix the first row and the first column, ( k ) is fixed, and the rest of the array is determined.But we need to ensure that this construction results in an array where all entries are between 1 and 2013. However, since we're working modulo 2013, the entries ( a_{i,j} ) are already given to be between 1 and 2013, so as long as the first row and first column are chosen appropriately, the rest will fall into place.Therefore, the number of such arrays is equal to the number of possible first rows and first columns, considering that ( a_{1,1} ) is shared.The first row has 2013 entries, each from 1 to 2013, so there are ( 2013^{2013} ) possibilities. Similarly, the first column has 2013 entries, but the first entry ( a_{1,1} ) is already chosen as part of the first row. So, the first column has ( 2013^{2012} ) possibilities (since ( a_{1,1} ) is fixed).Therefore, the total number of arrays is ( 2013^{2013} times 2013^{2012} = 2013^{4025} ).Wait, but I need to make sure that this construction actually works. Let me verify with a small example. Suppose we have a 2x2 array.Let me choose the first row as [1, 2] and the first column as [1, 3]. Then, ( a_{1,1} = 1 ), so ( k = -1 ).Then, ( a_{2,2} equiv a_{2,1} + a_{1,2} + k pmod{2013} ). So, ( a_{2,2} equiv 3 + 2 - 1 = 4 pmod{2013} ). So, ( a_{2,2} = 4 ).Similarly, ( a_{2,1} = 3 ), which is consistent with the first column.So, the array would be:1 23 4Now, let's see if we can perform operations to make all entries divisible by 2013. Since 2013 is larger than 4, this might not be the best example, but let's try.Wait, in this small example, 2013 is much larger than the entries, so making them divisible by 2013 would require adding or subtracting multiples of 2013, which isn't possible with the given operations since we can only add or subtract 1 per row or column.Hmm, maybe I need a different approach. Perhaps I should think in terms of the differences between rows and columns.Wait, going back to the original problem, since we're working modulo 2013, the operations of subtracting 1 from a row or adding 1 to a column are equivalent to adding 2012 or adding 1 modulo 2013, respectively.So, the key is that the operations allow us to adjust the entries in such a way that we can reach any residue modulo 2013, provided the system of congruences is consistent.From earlier, we saw that the array must satisfy ( a_{i,j} equiv a_{i,1} + a_{1,j} + k pmod{2013} ) for some constant ( k ). This is a necessary condition for the system to have a solution.Therefore, the number of such arrays is equal to the number of ways to choose the first row, first column, and ( k ), but since ( k ) is determined by ( a_{1,1} ), it's actually just the number of ways to choose the first row and first column.But wait, ( k ) is determined by ( a_{1,1} ), so once we choose the first row and first column, ( k ) is fixed. Therefore, the number of arrays is indeed ( 2013^{2013 + 2013 - 1} = 2013^{4025} ).This is because the first row has 2013 entries, the first column has 2013 entries, but they share the entry ( a_{1,1} ), so we have to subtract 1 to avoid double-counting. Therefore, the total number of degrees of freedom is ( 2013 + 2013 - 1 = 4025 ), each with 2013 choices.So, the final answer is ( 2013^{4025} ).

💡Alright, so I have this sequence problem here. It says that the sequence {a_n} satisfies a_4 = 1/8, and for any positive integer n, it satisfies two inequalities: a_{n+2} - a_n ≤ 3^n and a_{n+4} - a_n ≥ 10×3^n. I need to find a_{2016}. Hmm, okay, let me try to unpack this step by step.First, let me write down what I know:1. a_4 = 1/8.2. For any positive integer n, a_{n+2} - a_n ≤ 3^n.3. For any positive integer n, a_{n+4} - a_n ≥ 10×3^n.I need to find a_{2016}. That's a pretty large index, so I'm guessing there's a pattern or a recursive formula that I can use to express a_{2016} in terms of a_4.Let me think about the inequalities given. The first one says that the difference between a term two places ahead and the current term is at most 3^n. The second one says that the difference between a term four places ahead and the current term is at least 10×3^n.Maybe I can combine these two inequalities somehow. Let's see.If I take the first inequality, a_{n+2} - a_n ≤ 3^n, and then consider the term two places after that, which would be a_{n+4} - a_{n+2} ≤ 3^{n+2}. So, adding these two inequalities together, I get:(a_{n+2} - a_n) + (a_{n+4} - a_{n+2}) ≤ 3^n + 3^{n+2}Simplifying the left side, the a_{n+2} terms cancel out, so I get:a_{n+4} - a_n ≤ 3^n + 3^{n+2}Let me compute 3^n + 3^{n+2}:3^n + 3^{n+2} = 3^n + 9×3^n = 10×3^nSo, combining this with the second inequality given, which is a_{n+4} - a_n ≥ 10×3^n, I have:10×3^n ≤ a_{n+4} - a_n ≤ 10×3^nTherefore, the difference a_{n+4} - a_n must be exactly equal to 10×3^n. So, we have:a_{n+4} - a_n = 10×3^nThat's a key insight! This means that every four terms, the sequence increases by 10×3^n. So, starting from a_4, each subsequent term four steps ahead is 10×3^n more than the current term.Now, I need to find a_{2016}. Let's see how many steps of four are there from a_4 to a_{2016}.First, let me compute how many terms are between a_4 and a_{2016}. Since 2016 - 4 = 2012, that's 2012 terms apart. But since we're dealing with steps of four, I need to divide 2012 by 4 to find how many four-step intervals there are.2012 ÷ 4 = 503. So, there are 503 intervals of four terms each from a_4 to a_{2016}.Therefore, starting from a_4, each four-term step adds 10×3^n, where n starts at 4 and increases by 4 each time. Wait, no, actually, n is the index, so when we take a_{n+4} - a_n = 10×3^n, n is the starting index.So, for each step from a_4 to a_8, a_8 to a_{12}, and so on, up to a_{2012} to a_{2016}, each of these steps adds 10×3^n, where n is 4, 8, 12, ..., 2012.Therefore, the total increase from a_4 to a_{2016} is the sum of 10×3^4 + 10×3^8 + 10×3^{12} + ... + 10×3^{2012}.This is a geometric series where each term is 10×3^{4k} for k from 1 to 503, since 4×503 = 2012.Wait, let me check: starting from n=4, the exponents are 4, 8, 12, ..., 2012. So, that's 4×1, 4×2, 4×3, ..., 4×503. So, yes, the exponents are multiples of 4 up to 2012.So, the sum S = 10×(3^4 + 3^8 + 3^{12} + ... + 3^{2012}).This is a geometric series with first term a = 3^4 = 81, common ratio r = 3^4 = 81, and number of terms N = 503.The formula for the sum of a geometric series is S = a×(r^N - 1)/(r - 1).Plugging in the values:S = 81×(81^{503} - 1)/(81 - 1) = 81×(81^{503} - 1)/80.But remember, the total increase is 10×S, so:Total increase = 10×[81×(81^{503} - 1)/80] = (10/80)×81×(81^{503} - 1) = (1/8)×81×(81^{503} - 1).Simplify that:Total increase = (81/8)×(81^{503} - 1).Therefore, a_{2016} = a_4 + total increase = 1/8 + (81/8)×(81^{503} - 1).Let me compute that:First, factor out 1/8:a_{2016} = (1 + 81×(81^{503} - 1))/8.Compute the numerator:1 + 81×81^{503} - 81×1 = 1 - 81 + 81^{504} = (1 - 81) + 81^{504} = -80 + 81^{504}.So, numerator is 81^{504} - 80.Therefore, a_{2016} = (81^{504} - 80)/8.Let me double-check my steps to make sure I didn't make a mistake.1. Started with a_4 = 1/8.2. Derived that a_{n+4} - a_n = 10×3^n.3. Noted that from a_4 to a_{2016} is 503 steps of four terms each.4. Summed the series 10×(3^4 + 3^8 + ... + 3^{2012}).5. Recognized it's a geometric series with a=81, r=81, N=503.6. Applied the formula correctly, multiplied by 10, then by 1/8.7. Simplified the numerator correctly to 81^{504} - 80.Seems solid. I think that's the correct answer.**Final Answer**The value of ( a_{2016} ) is boxed{dfrac{81^{504} - 80}{8}}.

💡Okay, so I have this problem here about spherical coordinates, and I need to figure out what shape is described when the angle θ is constant at π/4. The options are Line, Circle, Plane, Sphere, Cylinder, or Cone. Hmm, let me think about this step by step.First, I remember that in spherical coordinates, there are three components: ρ, θ, and φ. ρ is the radius, which is the distance from the origin to the point. θ is the azimuthal angle, which is the angle in the xy-plane measured from the positive x-axis. φ is the polar angle, measured from the positive z-axis down to the point.So, if θ is fixed at π/4, that means every point we're considering has the same angle in the xy-plane. Specifically, π/4 radians is 45 degrees. So, in the xy-plane, all these points lie on a line that makes a 45-degree angle with the positive x-axis. But wait, in three dimensions, it's not just a line because ρ and φ can vary.Let me visualize this. If θ is fixed, it's like we're looking at all points that lie on a half-plane that starts from the origin and extends out in the direction of θ = π/4 in the xy-plane. But since ρ can be any positive value, this half-plane extends infinitely in all directions perpendicular to the x-y plane. So, it's not just a line or a circle; it's more like a flat surface that goes on forever.Wait, but is it a plane? A plane is a flat, two-dimensional surface that extends infinitely in all directions. In this case, since θ is fixed, all points lie on a specific plane that makes a 45-degree angle with the x-axis. This plane would include all points where the projection onto the xy-plane is at θ = π/4, but their height (z-coordinate) can vary because φ can change. So, yeah, that sounds like a plane.Let me double-check. If θ were fixed, say θ = 0, then all points would lie along the positive x-axis in the xy-plane, but since ρ and φ can vary, it's not just the x-axis; it's the entire plane that contains the x-axis and extends infinitely in all directions perpendicular to the y-axis. Similarly, for θ = π/4, it's a plane that makes a 45-degree angle with the x-axis in the xy-plane and extends infinitely in the z-direction.So, considering all that, the shape described by θ = π/4 in spherical coordinates is a plane. That means the correct answer should be (C) Plane.But just to be thorough, let's consider the other options:- Line: A line is one-dimensional and straight, but here we're talking about a two-dimensional surface, so it's not a line.- Circle: A circle is a one-dimensional curve, but again, we have a two-dimensional surface here.- Sphere: A sphere would require ρ to be constant, not θ.- Cylinder: A cylinder typically has a fixed radius in cylindrical coordinates, but here θ is fixed, which is different.- Cone: A cone would involve a relationship between ρ and φ, not just a fixed θ.So, yeah, none of the other options fit as well as a plane. Therefore, I'm confident that the answer is a plane.