💡First, I need to determine the elevation of Boris' mountain. Since Boris' mountain is 2,500 feet shorter than Hugo's mountain, I subtract 2,500 feet from Hugo's mountain elevation:10,000 feet (Hugo's mountain) - 2,500 feet = 7,500 feet (Boris' mountain)Next, I calculate how many feet Hugo climbed after climbing his mountain 3 times:10,000 feet x 3 climbs = 30,000 feetTo find out how many times Boris needs to climb his mountain to match Hugo's total climbed feet, I divide the total feet Hugo climbed by the elevation of Boris' mountain:30,000 feet (Hugo's total climbed feet) ÷ 7,500 feet (Boris' mountain) = 4 climbsTherefore, Boris would need to climb his mountain 4 times to have climbed the same number of feet as Hugo.

💡Okay, so I'm trying to figure out the missing number in this sequence: 4, 5, 14, 15, 24, 25, and then there's a missing number. Alright, let's see. First, I'll write down the numbers to get a better look: 4, 5, 14, 15, 24, 25, ?Hmm, I notice that the numbers are increasing, but not by the same amount each time. Maybe there's a pattern in how they increase. Let me calculate the differences between each pair of consecutive numbers.So, 5 minus 4 is 1. Then, 14 minus 5 is 9. Next, 15 minus 14 is 1 again. Then, 24 minus 15 is 9. And finally, 25 minus 24 is 1. Okay, so the differences are alternating between 1 and 9. It goes up by 1, then up by 9, then up by 1, then up by 9, and so on.If that's the case, then after 25, which was an increase of 1 from 24, the next increase should be by 9. So, 25 plus 9 would be 34. That seems to fit the pattern.But wait, let me double-check to make sure I'm not missing something. Maybe there's another pattern here. Let's see, are there any other ways to look at this sequence? For example, could it be related to multiples of some number or perhaps following a different mathematical rule?Looking at the numbers: 4, 5, 14, 15, 24, 25. I notice that 4 and 5 are consecutive numbers, then 14 and 15 are consecutive, and 24 and 25 are consecutive. So, it seems like there are pairs of consecutive numbers separated by larger jumps.Between 5 and 14, the jump is 9, and between 15 and 24, the jump is also 9. So, the pattern is that after every pair of consecutive numbers, there's a jump of 9 to get to the next pair. That makes sense with the differences I calculated earlier.So, following this pattern, after 25, the next number should be 25 plus 9, which is 34. That seems consistent.Is there any other way to interpret this sequence? Maybe looking at the numbers in terms of their digits or some other property? Let's see.4 is 4, 5 is 5, 14 is 1 and 4, 15 is 1 and 5, 24 is 2 and 4, 25 is 2 and 5. Hmm, I don't see a clear pattern in the digits themselves. It doesn't seem like they're following a specific rule based on their digits.What about if I consider the numbers in terms of their positions? Let's assign each number a position:1: 42: 53: 144: 155: 246: 257: ?Looking at this, maybe there's a pattern based on the position. For example, odd positions and even positions might have different rules.Looking at the odd positions: 1 is 4, 3 is 14, 5 is 24. So, 4, 14, 24. That seems like it's increasing by 10 each time. 4 plus 10 is 14, 14 plus 10 is 24. So, if that's the case, the next odd position, which would be position 7, should be 24 plus 10, which is 34.For the even positions: 2 is 5, 4 is 15, 6 is 25. So, 5, 15, 25. That's also increasing by 10 each time. 5 plus 10 is 15, 15 plus 10 is 25. So, the next even position, which would be position 8, should be 25 plus 10, which is 35.But in our sequence, we're only asked for the 7th number, which is an odd position, so that would be 34. That matches what I found earlier.So, both approaches—looking at the differences between consecutive numbers and looking at the pattern in odd and even positions—lead me to the same conclusion that the missing number is 34.I think that makes sense. The sequence alternates between adding 1 and adding 9, and when you look at the positions, the odd positions increase by 10 each time, and the even positions also increase by 10 each time. So, the missing number should be 34.

💡First, I note that the river's depth in mid-May is 5 feet.In mid-June, the river is 10 feet deeper than in mid-May. So, I add 10 feet to the mid-May depth to find the mid-June depth:5 feet + 10 feet = 15 feet.By mid-July, the river is three times deeper than in mid-June. To find the mid-July depth, I multiply the mid-June depth by 3:15 feet × 3 = 45 feet.Therefore, the river was 45 feet deep in mid-July.

💡Okay, so I have this problem here about determining which trajectory of point M isn't on a straight line. There are four options: A, B, C, and D. Each one describes a different condition involving distances from point M to certain lines or points. I need to figure out which one doesn't result in a straight line path for M.Let me start by understanding each option one by one.**Option A:** The sum of the distances from point M to the lines (4x + 3y - 5 = 0) and (4x + 3y + 10 = 0) is 3.Hmm, okay. So, these are two parallel lines because they have the same coefficients for x and y. The distance between these two lines can be calculated using the formula for the distance between two parallel lines: ( frac{|c_2 - c_1|}{sqrt{a^2 + b^2}} ). Here, (c_1 = -5) and (c_2 = 10), so the distance is ( frac{|10 - (-5)|}{sqrt{4^2 + 3^2}} = frac{15}{5} = 3 ).So, the distance between these two lines is 3. Now, the condition is that the sum of the distances from M to each line is 3. If I think about it, if you have two parallel lines, any point between them will have the sum of distances to both lines equal to the distance between the lines. So, in this case, the sum is exactly 3, which is the distance between the lines. That means the trajectory of M is the region between these two lines. Wait, but the region between two parallel lines is not a straight line; it's like a strip. So, this trajectory isn't a straight line. That might be the answer, but let me check the others just to be sure.**Option B:** The sum of the distances from point M to the lines passing through points (1, 0) and (-1, 0) is 2.Alright, so these two points are (1,0) and (-1,0). The lines passing through these points... Wait, hold on. If it's lines passing through these points, there are infinitely many lines passing through each point. But the wording says "the lines passing through points (1, 0) and (-1, 0)". Hmm, maybe it's the two lines that pass through each of these points? Or is it the line connecting these two points?Wait, actually, if you have two points, (1,0) and (-1,0), the line passing through both is the x-axis, right? Because both points lie on the x-axis. So, if the sum of the distances from M to both lines is 2, but if both lines are the same line (the x-axis), then the distance from M to the x-axis is just the absolute y-coordinate. So, the sum would be twice the distance, which would be 2|y|. Setting that equal to 2 gives |y| = 1. So, the trajectory is the lines y = 1 and y = -1. These are straight lines. So, this trajectory is on straight lines. So, B is not the answer.Wait, hold on. Maybe I misinterpreted the lines. The problem says "the lines passing through points (1, 0) and (-1, 0)". So, each line passes through one of the points? Or both? If it's both, then it's the x-axis. If it's each, then there are two lines: one passing through (1,0) and another through (-1,0). But without more information, it's ambiguous. But given the context, it's more likely referring to the line connecting (1,0) and (-1,0), which is the x-axis. So, sum of distances to the x-axis is 2, which as I said, gives y = 1 and y = -1. So, definitely straight lines. So, B is not the answer.**Option C:** The difference in distances from point M to the lines passing through points (0, 2) and (0, -2) is 4.Alright, similar to B, but now the difference in distances. The points are (0,2) and (0,-2). So, the lines passing through these points. Again, if it's the lines passing through each point, then we have two vertical lines at x=0? Wait, no, (0,2) and (0,-2) are both on the y-axis. So, the lines passing through these points would be the y-axis. But if it's the lines passing through each point, then again, infinitely many lines. But the problem says "the lines passing through points (0, 2) and (0, -2)", which is the y-axis. So, the difference in distances from M to the y-axis is 4.Wait, the distance from a point to the y-axis is just the absolute x-coordinate. So, the difference would be |x| - |x|, which is zero, or maybe | |x| - |x| |, which is still zero. That doesn't make sense. Maybe I'm misinterpreting.Alternatively, maybe the lines are not the y-axis but lines passing through each of these points. So, for (0,2), the line could be any line passing through (0,2), and similarly for (0,-2). But without more information, it's unclear. Alternatively, maybe it's the two lines that pass through each of these points, but are not necessarily the same line.Wait, maybe it's the two lines that are the vertical lines x=0, but that's just the y-axis. Hmm, I'm confused. Alternatively, maybe it's the lines connecting (0,2) and (0,-2), which is the y-axis again.Wait, perhaps it's the two lines that are the perpendicular bisectors or something else. Alternatively, maybe it's the two lines that are each at a distance such that the difference is 4.Wait, maybe it's better to think in terms of hyperbolas. The difference of distances from two points is constant, which is a hyperbola. But in this case, it's the difference in distances to two lines, not two points. Hmm, so if it's the difference in distances to two lines, what does that trajectory look like?If the two lines are parallel, then the difference in distances would be constant along certain lines. If they're not parallel, it might be more complicated. But in this case, the lines passing through (0,2) and (0,-2). If the lines are the y-axis, then the difference in distance is zero, which doesn't make sense. Alternatively, if the lines are something else.Wait, maybe the lines are the lines connecting (0,2) and (0,-2) to some other points? No, the problem says "the lines passing through points (0,2) and (0,-2)". So, it's lines that pass through these two points. But a single line can pass through both (0,2) and (0,-2), which is the y-axis. So, again, the distance from M to the y-axis is |x|, so the difference would be |x| - |x| = 0, which is not 4. So, that doesn't make sense.Alternatively, maybe it's two different lines, each passing through one of the points. For example, one line passes through (0,2) and another passes through (0,-2). But without knowing the slopes, it's impossible to determine. Maybe the lines are horizontal or vertical.Wait, if the lines are horizontal, then the lines would be y=2 and y=-2. So, the distance from M(x,y) to y=2 is |y - 2|, and to y=-2 is |y + 2|. The difference would be | |y - 2| - |y + 2| |. Let's compute that.If y >= 2, then |y - 2| = y - 2 and |y + 2| = y + 2. So, the difference is (y - 2) - (y + 2) = -4. Absolute value is 4.If -2 <= y <= 2, then |y - 2| = 2 - y and |y + 2| = y + 2. So, the difference is (2 - y) - (y + 2) = -2y. The absolute value is | -2y | = 2|y|.If y <= -2, then |y - 2| = -(y - 2) = -y + 2 and |y + 2| = -(y + 2) = -y - 2. So, the difference is (-y + 2) - (-y - 2) = 4. Absolute value is 4.So, the difference is 4 when y >= 2 or y <= -2, and 2|y| otherwise. So, if the difference is 4, then y >= 2 or y <= -2. So, the trajectory is the regions y >= 2 and y <= -2. But that's not a straight line; it's two half-planes. Wait, but the problem says "the difference in distances... is 4". So, does that mean the absolute difference is 4? If so, then yes, it's y >= 2 or y <= -2, which are regions, not lines. So, is that a straight line? No, it's regions above and below. So, maybe C is not a straight line either.Wait, but in the initial analysis, I thought it might be a hyperbola, but it turned out to be regions. Hmm, maybe I need to reconsider.Alternatively, if the lines are not horizontal, but something else. For example, if the lines are the lines passing through (0,2) and (0,-2) with some slope. Let's say the lines are y = mx + 2 and y = mx - 2. Then, the distance from M(x,y) to each line can be calculated, and the difference set to 4. That might result in a hyperbola or something else.But without knowing the slope, it's hard to say. Maybe the problem assumes the lines are the y-axis, but that didn't make sense earlier. Alternatively, maybe the lines are the lines connecting (0,2) and (0,-2) to some other points, but the problem doesn't specify.Wait, maybe the lines are the lines passing through each of the points (0,2) and (0,-2), but not necessarily the same line. So, for example, one line passes through (0,2) and another passes through (0,-2), but they could be any lines. But without more information, it's impossible to determine. So, maybe the problem is assuming that the lines are the y-axis, but as I saw earlier, that leads to a difference of zero, which isn't 4.Alternatively, maybe the lines are the lines perpendicular to the y-axis, i.e., horizontal lines y=2 and y=-2. Then, as I calculated earlier, the difference in distances is 4 when y >= 2 or y <= -2. So, the trajectory is the regions above y=2 and below y=-2. So, that's not a straight line, it's two regions. So, maybe C is also not a straight line.Wait, but in the initial problem, the options are A, B, C, D. So, if both A and C are not straight lines, but the question is asking which one is not on a straight line, implying only one answer. So, maybe I made a mistake in analyzing C.Wait, let me think again. If the lines are y=2 and y=-2, then the difference in distances is 4 when y >= 2 or y <= -2. So, the trajectory is the union of y >= 2 and y <= -2. That's two regions, not a straight line. So, C is not a straight line. Similarly, A is a region between two lines, which is also not a straight line. So, both A and C are not straight lines. But the problem says "which of the following trajectories... is not on a straight line". So, maybe only one of them is not a straight line, and the others are. So, perhaps I'm misunderstanding something.Wait, let me go back to A. The sum of distances from M to two parallel lines is equal to the distance between the lines. So, in that case, the set of points M where the sum of distances is equal to the distance between the lines is the region between the two lines. So, that's a strip, not a straight line. So, A is not a straight line.In B, the sum of distances to the x-axis is 2, which gives y=1 and y=-1, which are straight lines. So, B is a straight line.In C, if the lines are y=2 and y=-2, then the difference in distances is 4 when y >= 2 or y <= -2, which are regions, not lines. So, C is not a straight line.In D, the distance from M to (2,3) is equal to its distance to the line 2x - y -1 =0. That's the definition of a parabola, right? The set of points equidistant from a point and a line is a parabola, which is a curve, not a straight line. So, D is not a straight line.Wait, so according to this, A, C, and D are not straight lines, while B is. But the problem is asking which one is not on a straight line, implying only one answer. So, perhaps I made a mistake in interpreting C.Wait, let me think again about C. The difference in distances from M to the lines passing through (0,2) and (0,-2) is 4. If the lines are the y-axis, then the difference is zero, which isn't 4. If the lines are horizontal lines y=2 and y=-2, then the difference is 4 when y >= 2 or y <= -2, which are regions, not lines. Alternatively, if the lines are something else.Wait, maybe the lines are the lines connecting (0,2) and (0,-2) to some other points, but the problem doesn't specify. Alternatively, maybe the lines are the lines passing through each of the points (0,2) and (0,-2) with some slope, say, 1. So, lines y = x + 2 and y = x - 2. Then, the distance from M(x,y) to each line is |x - y + 2| / sqrt(2) and |x - y - 2| / sqrt(2). The difference would be | |x - y + 2| - |x - y - 2| | / sqrt(2) = 4.Let me compute this. Let me set z = x - y. Then, the expression becomes | |z + 2| - |z - 2| | / sqrt(2) = 4.So, | |z + 2| - |z - 2| | = 4 sqrt(2).Let me consider different cases for z.Case 1: z >= 2.Then, |z + 2| = z + 2, |z - 2| = z - 2.So, | (z + 2) - (z - 2) | = |4| = 4.So, 4 = 4 sqrt(2). That's not possible because 4 ≠ 4 sqrt(2).Case 2: -2 <= z <= 2.Then, |z + 2| = z + 2, |z - 2| = 2 - z.So, | (z + 2) - (2 - z) | = |2z|.Set |2z| = 4 sqrt(2). So, |z| = 2 sqrt(2). But in this case, z is between -2 and 2, so |z| = 2 sqrt(2) is not possible because 2 sqrt(2) ≈ 2.828 > 2.Case 3: z <= -2.Then, |z + 2| = -(z + 2), |z - 2| = -(z - 2).So, | (-z - 2) - (-z + 2) | = | -4 | = 4.Again, 4 = 4 sqrt(2), which is not possible.So, in this case, there are no solutions. So, maybe the lines are not y = x + 2 and y = x - 2.Alternatively, maybe the lines are vertical lines x=0, but that didn't work earlier.Alternatively, maybe the lines are horizontal lines y=2 and y=-2, which gave us regions y >= 2 and y <= -2. So, in that case, the trajectory is two regions, not a straight line.Alternatively, maybe the lines are something else, but without more information, it's hard to say. So, perhaps the problem assumes that the lines are the y-axis, but that leads to a difference of zero, which isn't 4. Alternatively, maybe the lines are the lines connecting (0,2) and (0,-2) to some other points, but the problem doesn't specify.Wait, maybe the lines are the lines passing through (0,2) and (0,-2) with some slope, but the problem doesn't specify, so it's ambiguous. Maybe the intended answer is that C is a straight line, but I'm not sure.Wait, let me think about the difference of distances. If the two lines are parallel, then the difference in distances would be constant along certain lines. For example, if the lines are horizontal, y=2 and y=-2, then the difference in distances is 4 when y >= 2 or y <= -2, which are regions, not lines. So, in that case, it's not a straight line.Alternatively, if the lines are not parallel, then the difference in distances could result in a hyperbola or something else, which is a curve, not a straight line.So, perhaps C is not a straight line either.Wait, but the problem is asking which one is not on a straight line, implying only one answer. So, maybe I made a mistake in analyzing A or C.Wait, let me go back to A. The sum of distances from M to two parallel lines is equal to the distance between the lines. So, the set of points M where the sum of distances is equal to the distance between the lines is the region between the two lines. So, that's a strip, not a straight line. So, A is not a straight line.In B, the sum of distances to the x-axis is 2, which gives y=1 and y=-1, which are straight lines.In C, if the lines are y=2 and y=-2, then the difference in distances is 4 when y >= 2 or y <= -2, which are regions, not lines. So, C is not a straight line.In D, the distance from M to (2,3) is equal to its distance to the line 2x - y -1 =0. That's a parabola, which is a curve, not a straight line.So, according to this, A, C, and D are not straight lines, while B is. But the problem is asking which one is not on a straight line, implying only one answer. So, perhaps the intended answer is A, and I'm overcomplicating C and D.Wait, let me think again. Maybe in C, the lines are not horizontal, but something else, and the difference in distances results in a straight line.Wait, if the lines are not parallel, then the difference in distances could result in a straight line. For example, if the lines intersect, then the set of points where the difference in distances is constant could be a straight line.Wait, let me consider two intersecting lines. Suppose the lines are y = 2 and y = -2, which are parallel. But if they intersect, say, at some point, then the difference in distances could be a straight line.Wait, but in this case, the lines passing through (0,2) and (0,-2) would be the y-axis if they intersect at (0,0). Wait, no, the lines passing through (0,2) and (0,-2) would be the y-axis if they are the same line, but they are two different points. So, maybe the lines are the lines connecting (0,2) and (0,-2) to some other points, but the problem doesn't specify.Wait, maybe the lines are the lines passing through (0,2) and (0,-2) with some slope, say, m. So, the lines would be y = m x + 2 and y = m x - 2. Then, the distance from M(x,y) to each line is |m x - y + 2| / sqrt(m^2 + 1) and |m x - y - 2| / sqrt(m^2 + 1). The difference would be | |m x - y + 2| - |m x - y - 2| | / sqrt(m^2 + 1) = 4.Let me set z = m x - y. Then, the expression becomes | |z + 2| - |z - 2| | / sqrt(m^2 + 1) = 4.So, | |z + 2| - |z - 2| | = 4 sqrt(m^2 + 1).Let me consider different cases for z.Case 1: z >= 2.Then, |z + 2| = z + 2, |z - 2| = z - 2.So, | (z + 2) - (z - 2) | = |4| = 4.So, 4 = 4 sqrt(m^2 + 1).Divide both sides by 4: 1 = sqrt(m^2 + 1).Square both sides: 1 = m^2 + 1.So, m^2 = 0, which implies m = 0.So, if m = 0, then the lines are y = 2 and y = -2, which are horizontal lines. Then, as I saw earlier, the difference in distances is 4 when y >= 2 or y <= -2, which are regions, not lines.Case 2: -2 <= z <= 2.Then, |z + 2| = z + 2, |z - 2| = 2 - z.So, | (z + 2) - (2 - z) | = |2z|.Set |2z| = 4 sqrt(m^2 + 1).So, |z| = 2 sqrt(m^2 + 1).But in this case, z is between -2 and 2, so 2 sqrt(m^2 + 1) must be <= 2.So, sqrt(m^2 + 1) <= 1.Which implies m^2 + 1 <= 1.So, m^2 <= 0, which implies m = 0.But if m = 0, then as before, the lines are y=2 and y=-2, and the difference in distances is 4 when y >= 2 or y <= -2.Case 3: z <= -2.Then, |z + 2| = -(z + 2), |z - 2| = -(z - 2).So, | (-z - 2) - (-z + 2) | = | -4 | = 4.So, 4 = 4 sqrt(m^2 + 1).Again, same as case 1, m = 0.So, in all cases, m must be 0, which means the lines are horizontal, y=2 and y=-2, and the difference in distances is 4 when y >= 2 or y <= -2, which are regions, not lines.Therefore, C is not a straight line.Similarly, D is a parabola, which is a curve, not a straight line.So, in conclusion, A, C, and D are not straight lines, while B is. But the problem is asking which one is not on a straight line, implying only one answer. So, perhaps the intended answer is A, and the others are intended to be straight lines, but my analysis shows that C and D are also not straight lines. Maybe I'm missing something.Wait, let me think about D again. The distance from M to (2,3) is equal to its distance to the line 2x - y -1 =0. That's the definition of a parabola, which is a curve, not a straight line. So, D is not a straight line.Wait, but maybe in some special case, it could be a straight line. For example, if the point and the line are arranged in a certain way, the set of points equidistant could be a straight line. But generally, it's a parabola. So, unless the point lies on the line, which it doesn't in this case, because (2,3) is not on 2x - y -1 =0 (since 2*2 - 3 -1 = 4 -3 -1 = 0, wait, actually, it is on the line! Wait, 2*2 - 3 -1 = 4 -3 -1 = 0. So, (2,3) lies on the line 2x - y -1 =0.Wait, that's interesting. So, if the point lies on the line, then the set of points equidistant from the point and the line is the line itself. Because any point on the line is equidistant to the line and the point (since the distance to the line is zero, and the distance to the point is the distance from the point to the line, which is also zero for points on the line). Wait, no, that's not quite right.Wait, if the point is on the line, then the set of points equidistant from the point and the line would be the line itself, because for any point on the line, the distance to the line is zero, and the distance to the point is the distance between the two points. But that's not necessarily zero unless the point is the same as the given point.Wait, no, actually, if the point is on the line, then the set of points equidistant from the point and the line is the line itself. Because for any point on the line, the distance to the line is zero, and the distance to the point is the distance between the two points. But that's not necessarily equal unless the point is the same as the given point.Wait, maybe I'm overcomplicating. Let me think about it mathematically.Let me denote the point as P(2,3) and the line as L: 2x - y -1 =0.Since P lies on L, because 2*2 - 3 -1 = 0.Now, the set of points M such that distance(M, P) = distance(M, L).But since P is on L, for any point M on L, distance(M, L) = 0, and distance(M, P) is the distance between M and P. So, unless M = P, distance(M, P) is not zero. So, the only point where distance(M, P) = distance(M, L) is P itself.Wait, that can't be right. Let me think again.Wait, no, if M is on L, then distance(M, L) = 0, and distance(M, P) is the distance between M and P. So, unless M = P, distance(M, P) is not zero. So, the only point where distance(M, P) = distance(M, L) is P itself.But that's just a single point, not a line.Wait, but if M is not on L, then distance(M, L) is the perpendicular distance, and distance(M, P) is the Euclidean distance. So, setting them equal would give a parabola, but since P is on L, maybe it's a different curve.Wait, let me write the equation.Let M(x,y). Then, distance(M, P) = sqrt( (x - 2)^2 + (y - 3)^2 ).Distance(M, L) = |2x - y -1| / sqrt(2^2 + (-1)^2) = |2x - y -1| / sqrt(5).Setting them equal:sqrt( (x - 2)^2 + (y - 3)^2 ) = |2x - y -1| / sqrt(5).Square both sides:( (x - 2)^2 + (y - 3)^2 ) = ( (2x - y -1)^2 ) / 5.Multiply both sides by 5:5( (x - 2)^2 + (y - 3)^2 ) = (2x - y -1)^2.Expand both sides:Left side: 5(x^2 -4x +4 + y^2 -6y +9) = 5x^2 -20x +20 +5y^2 -30y +45 = 5x^2 +5y^2 -20x -30y +65.Right side: (2x - y -1)^2 = 4x^2 -4xy + y^2 -4x + 2y +1.So, set left = right:5x^2 +5y^2 -20x -30y +65 = 4x^2 -4xy + y^2 -4x + 2y +1.Bring all terms to left:5x^2 +5y^2 -20x -30y +65 -4x^2 +4xy - y^2 +4x -2y -1 = 0.Simplify:(5x^2 -4x^2) + (5y^2 - y^2) + (-20x +4x) + (-30y -2y) + (65 -1) +4xy = 0.So:x^2 +4y^2 -16x -32y +64 +4xy = 0.Hmm, this is a quadratic equation. Let me see if it's a parabola or something else.The general form is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.Here, A=1, B=4, C=4, D=-16, E=-32, F=64.To determine the type of conic, compute the discriminant: B^2 - 4AC = 16 - 16 = 0.Since the discriminant is zero, it's a parabola.So, D is a parabola, which is a curve, not a straight line.Therefore, D is not a straight line.So, in conclusion, A, C, and D are not straight lines, while B is. But the problem is asking which one is not on a straight line, implying only one answer. So, perhaps the intended answer is A, and the others are intended to be straight lines, but my analysis shows that C and D are also not straight lines. Maybe I'm missing something.Wait, let me think again about C. If the lines are the y-axis, then the difference in distances is zero, which isn't 4. If the lines are horizontal, y=2 and y=-2, then the difference in distances is 4 when y >= 2 or y <= -2, which are regions, not lines. So, C is not a straight line.Similarly, D is a parabola, which is a curve, not a straight line.So, perhaps the problem is designed such that only A is not a straight line, but my analysis shows that C and D are also not straight lines. Maybe the intended answer is A, and the others are intended to be straight lines, but my analysis shows that C and D are also not straight lines. Maybe I'm overcomplicating.Wait, let me think about B again. The sum of the distances from M to the lines passing through (1,0) and (-1,0) is 2. If the lines are the x-axis, then the sum is 2|y| = 2, so |y|=1, which are straight lines y=1 and y=-1. So, B is a straight line.In A, the sum of distances to two parallel lines is equal to the distance between them, which is a region between the lines, not a straight line.In C, the difference in distances to two lines is 4, which, if the lines are horizontal, results in regions y >= 2 and y <= -2, not straight lines.In D, the set of points equidistant from a point and a line is a parabola, which is a curve, not a straight line.So, according to this, A, C, and D are not straight lines, while B is. But the problem is asking which one is not on a straight line, implying only one answer. So, perhaps the intended answer is A, and the others are intended to be straight lines, but my analysis shows that C and D are also not straight lines. Maybe the problem is designed such that only A is not a straight line, and the others are.Alternatively, maybe I'm misinterpreting the problem. Let me read it again."The sum of the distances from point M to the lines passing through points (1, 0) and (-1, 0) is 2."Wait, maybe the lines are not the x-axis, but two different lines passing through each of the points. For example, one line passes through (1,0) and another passes through (-1,0), but they are not the same line. So, for example, one line could be y = m(x -1) and another line y = m(x +1). Then, the sum of distances from M(x,y) to these two lines is 2.But without knowing the slope m, it's hard to determine. Alternatively, if the lines are perpendicular to the x-axis, i.e., vertical lines x=1 and x=-1, then the distance from M(x,y) to each line is |x -1| and |x +1|. The sum would be |x -1| + |x +1|.If x >=1, then sum = (x -1) + (x +1) = 2x.Set 2x = 2 => x=1.If -1 <= x <=1, sum = (1 -x) + (x +1) = 2.So, for all x between -1 and 1, the sum is 2.If x <= -1, sum = -(x -1) - (x +1) = -2x.Set -2x = 2 => x = -1.So, the trajectory is the region between x=-1 and x=1, which is a vertical strip, not a straight line.Wait, that's different from my earlier analysis. So, if the lines are vertical lines x=1 and x=-1, then the sum of distances is 2 for all points between x=-1 and x=1. So, that's a vertical strip, not a straight line.But earlier, I thought the lines were the x-axis, but if the lines are vertical, then it's a different result.So, the problem says "the lines passing through points (1, 0) and (-1, 0)". So, if the lines are vertical lines passing through these points, then x=1 and x=-1, and the sum of distances is 2 for the region between them.Alternatively, if the lines are the x-axis, then the sum is 2|y|, giving y=1 and y=-1.So, which interpretation is correct? The problem says "the lines passing through points (1, 0) and (-1, 0)". So, it could be either the x-axis or two vertical lines. But since both points lie on the x-axis, the line passing through both is the x-axis. So, the sum of distances to the x-axis is 2|y| = 2, giving y=1 and y=-1, which are straight lines.Alternatively, if it's two lines passing through each point separately, then it could be vertical lines x=1 and x=-1, giving a vertical strip. But the problem says "the lines passing through points (1, 0) and (-1, 0)", which is singular "lines", plural. So, it's two lines, each passing through one of the points. So, it could be two vertical lines x=1 and x=-1.So, in that case, the sum of distances is |x -1| + |x +1|, which is 2 for -1 <= x <=1, which is a vertical strip, not a straight line.Wait, so depending on the interpretation, B could be a straight line or a vertical strip. But the problem is asking which trajectory is not on a straight line. So, if B is a vertical strip, then it's not a straight line. But if it's the x-axis, then it's straight lines.So, this is ambiguous. But given that the problem is asking for which one is not on a straight line, and considering that A, C, and D are not straight lines, while B could be either, depending on interpretation, perhaps the intended answer is A.But I'm not sure. Maybe the problem assumes that the lines are the x-axis, making B a straight line, and A, C, D not straight lines. But the problem is asking for which one is not on a straight line, implying only one answer. So, perhaps the intended answer is A.Alternatively, maybe the problem is designed such that only A is not a straight line, and the others are. But my analysis shows that C and D are also not straight lines. Maybe I'm overcomplicating.In conclusion, based on the analysis, A, C, and D are not straight lines, while B is. But since the problem is asking for which one is not on a straight line, and considering the options, the most straightforward answer is A.

💡First, I need to find a positive integer that is divisible by 21 and has a square root between 30 and 30.5.To determine the range of possible integers, I'll calculate the squares of 30 and 30.5:- (30^2 = 900)- (30.5^2 = 930.25)This means the integer must be between 900 and approximately 930.Next, I'll identify the multiples of 21 within this range:- Multiples of 21 between 900 and 930 are 903 and 924.Finally, I'll check the square roots of these multiples to ensure they fall within the specified range:- (sqrt{903} approx 30.05), which is within the range.- (sqrt{924} approx 30.40), which is also within the range.Both 903 and 924 are valid solutions. However, since the problem asks for a single integer, I'll choose the smallest one that satisfies all conditions.

💡Okay, so I have this problem where I'm given two matrices, M and N, and the inverse of their product MN. I need to find the matrix M. Let me try to break this down step by step.First, let me write down what I know:Matrix M is:[ M = begin{bmatrix} a & b c & d end{bmatrix} ]Matrix N is:[ N = begin{bmatrix} 1 & 0 0 & frac{1}{2} end{bmatrix} ]And the inverse of the product MN is:[ (MN)^{-1} = begin{bmatrix} frac{1}{4} & 0 0 & 2 end{bmatrix} ]I need to find the values of a, b, c, and d in matrix M. Hmm, okay. Let me recall some properties of matrix inverses and multiplication.I remember that if two matrices are invertible, then the inverse of their product is the product of their inverses in reverse order. So, ((MN)^{-1} = N^{-1}M^{-1}). That might be useful.But wait, in this problem, I don't know M, so maybe I can find MN first and then figure out M from that.Since ((MN)^{-1}) is given, I can find MN by taking the inverse of ((MN)^{-1}). That makes sense because the inverse of an inverse is the original matrix.So, let me compute MN by taking the inverse of ((MN)^{-1}):Given:[ (MN)^{-1} = begin{bmatrix} frac{1}{4} & 0 0 & 2 end{bmatrix} ]To find MN, I need to compute the inverse of this matrix. For a diagonal matrix, the inverse is just the reciprocal of each diagonal element.So, the inverse of ((MN)^{-1}) is:[ MN = begin{bmatrix} 4 & 0 0 & frac{1}{2} end{bmatrix} ]Okay, so now I have MN:[ MN = begin{bmatrix} 4 & 0 0 & frac{1}{2} end{bmatrix} ]Now, I need to find M. Since MN is known and N is given, I can find M by multiplying MN by the inverse of N. That is, M = MN * N^{-1}.Let me find N^{-1} first. Since N is a diagonal matrix, its inverse is also diagonal with the reciprocals of the diagonal elements.Given:[ N = begin{bmatrix} 1 & 0 0 & frac{1}{2} end{bmatrix} ]So, the inverse of N is:[ N^{-1} = begin{bmatrix} 1 & 0 0 & 2 end{bmatrix} ]Now, I can compute M by multiplying MN and N^{-1}:[ M = MN * N^{-1} ][ M = begin{bmatrix} 4 & 0 0 & frac{1}{2} end{bmatrix} * begin{bmatrix} 1 & 0 0 & 2 end{bmatrix} ]Let me perform this matrix multiplication step by step.First, the element in the first row and first column of M is:[ 4 * 1 + 0 * 0 = 4 ]The element in the first row and second column is:[ 4 * 0 + 0 * 2 = 0 ]The element in the second row and first column is:[ 0 * 1 + frac{1}{2} * 0 = 0 ]The element in the second row and second column is:[ 0 * 0 + frac{1}{2} * 2 = 1 ]So, putting it all together, matrix M is:[ M = begin{bmatrix} 4 & 0 0 & 1 end{bmatrix} ]Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting with MN:[ MN = begin{bmatrix} 4 & 0 0 & frac{1}{2} end{bmatrix} ]Multiplying by N^{-1}:[ N^{-1} = begin{bmatrix} 1 & 0 0 & 2 end{bmatrix} ]Multiplying MN and N^{-1}:- First row, first column: 4*1 + 0*0 = 4- First row, second column: 4*0 + 0*2 = 0- Second row, first column: 0*1 + (1/2)*0 = 0- Second row, second column: 0*0 + (1/2)*2 = 1Yes, that seems correct. So, matrix M is indeed:[ begin{bmatrix} 4 & 0 0 & 1 end{bmatrix} ]I think that makes sense because when you multiply M and N, you get MN as calculated, and taking the inverse gives the provided ((MN)^{-1}). So, everything checks out.Just to be thorough, let me verify by computing MN with the found M and given N.Given:[ M = begin{bmatrix} 4 & 0 0 & 1 end{bmatrix} ][ N = begin{bmatrix} 1 & 0 0 & frac{1}{2} end{bmatrix} ]Multiplying M and N:[ MN = begin{bmatrix} 4*1 + 0*0 & 4*0 + 0*(1/2) 0*1 + 1*0 & 0*0 + 1*(1/2) end{bmatrix} ][ MN = begin{bmatrix} 4 & 0 0 & frac{1}{2} end{bmatrix} ]Which matches what we had earlier. Then, taking the inverse of MN:[ (MN)^{-1} = begin{bmatrix} 1/4 & 0 0 & 2 end{bmatrix} ]Which is exactly the given ((MN)^{-1}). So, everything is consistent.Therefore, I'm confident that the matrix M is:[ begin{bmatrix} 4 & 0 0 & 1 end{bmatrix} ]