💡Okay, so I have this problem with four statements about lines and planes in space, and I need to figure out how many of them are correct. Let me go through each statement one by one and think it through.**Statement 1:** If ( m perp alpha ) and ( n parallel alpha ), then ( m perp n ).Alright, so ( m ) is perpendicular to plane ( alpha ), which means ( m ) is a line that forms a right angle with every line in ( alpha ) that it intersects. Now, ( n ) is parallel to ( alpha ). If a line is parallel to a plane, it means that the line doesn't intersect the plane and lies in a direction that's parallel to some line within the plane.Since ( m ) is perpendicular to the entire plane ( alpha ), it should be perpendicular to every line that lies on ( alpha ). But ( n ) is parallel to ( alpha ), so it's like ( n ) is running alongside ( alpha ) without touching it. Does that mean ( m ) is also perpendicular to ( n )?Hmm, I think yes. Because if ( n ) is parallel to ( alpha ), then ( n ) must be parallel to some line in ( alpha ). Since ( m ) is perpendicular to all lines in ( alpha ), it should also be perpendicular to any line parallel to ( alpha ). So, I think Statement 1 is correct.Wait, but actually, no. If ( n ) is parallel to ( alpha ), it doesn't necessarily mean that ( n ) is parallel to every line in ( alpha ). It just means ( n ) is parallel to the plane, so it's parallel to some direction in the plane. But ( m ) is perpendicular to the entire plane, so it should be perpendicular to any direction in the plane, including the one that ( n ) is parallel to. So, yes, ( m ) should be perpendicular to ( n ). So Statement 1 is correct.**Statement 2:** If ( m parallel n ) and ( n parallel alpha ), then ( m parallel alpha ).Okay, so if line ( m ) is parallel to line ( n ), and line ( n ) is parallel to plane ( alpha ), does that mean ( m ) is also parallel to ( alpha )?Well, if ( n ) is parallel to ( alpha ), it means ( n ) doesn't intersect ( alpha ) and lies in a direction parallel to ( alpha ). If ( m ) is parallel to ( n ), then ( m ) should also lie in the same direction, right? So, ( m ) should also be parallel to ( alpha ).But wait, could ( m ) be lying on ( alpha )? If ( m ) is parallel to ( n ) and ( n ) is parallel to ( alpha ), could ( m ) be on ( alpha )? Hmm, if ( m ) is on ( alpha ), then it's not parallel to ( alpha ); it's lying on it. So, does the statement allow for ( m ) to be on ( alpha )?The statement says ( m parallel n ) and ( n parallel alpha ). If ( m ) is on ( alpha ), then ( m ) is not parallel to ( alpha ); it's actually intersecting ( alpha ) everywhere along itself. So, is the statement saying that ( m ) must be parallel to ( alpha ), or could it also be lying on ( alpha )?I think the statement is saying that ( m ) is parallel to ( alpha ), but if ( m ) is on ( alpha ), it's not parallel. So, is the conclusion that ( m ) is parallel to ( alpha ) necessarily true?Wait, no. Because if ( m ) is parallel to ( n ), and ( n ) is parallel to ( alpha ), then ( m ) could either be parallel to ( alpha ) or lie on ( alpha ). So, the conclusion that ( m parallel alpha ) isn't necessarily always true because ( m ) could be on ( alpha ).Therefore, Statement 2 is incorrect because ( m ) might not be parallel to ( alpha ); it could lie on ( alpha ).Wait, but hold on. If ( m ) is parallel to ( n ), and ( n ) is parallel to ( alpha ), does that mean ( m ) can't lie on ( alpha )? Because if ( m ) is on ( alpha ), then ( m ) can't be parallel to ( n ) if ( n ) is parallel to ( alpha ). Hmm, I'm getting confused.Let me think again. If ( n parallel alpha ), then ( n ) doesn't lie on ( alpha ) and doesn't intersect ( alpha ). If ( m parallel n ), then ( m ) is also not intersecting ( n ). But ( m ) could be on ( alpha ) or parallel to ( alpha ). If ( m ) is on ( alpha ), then ( m ) is not parallel to ( alpha ); it's lying on it. So, is the statement claiming that ( m ) must be parallel to ( alpha ), which isn't necessarily the case because ( m ) could be on ( alpha )?Wait, but if ( m ) is on ( alpha ), is ( m parallel n ) possible? Because ( n ) is parallel to ( alpha ), so ( n ) doesn't intersect ( alpha ). If ( m ) is on ( alpha ), then ( m ) and ( n ) can't intersect, but they can be parallel. So, ( m ) can be on ( alpha ) and still be parallel to ( n ). Therefore, ( m ) could either be on ( alpha ) or parallel to ( alpha ). So, the conclusion that ( m parallel alpha ) isn't necessarily always true because ( m ) might be on ( alpha ).Therefore, Statement 2 is incorrect.Wait, but I'm not entirely sure. Maybe in 3D space, if ( m parallel n ) and ( n parallel alpha ), then ( m ) must be parallel to ( alpha ) or lie on ( alpha ). But the statement says "then ( m parallel alpha )", which might not always be the case because ( m ) could lie on ( alpha ). So, the conclusion isn't necessarily always true. Therefore, Statement 2 is incorrect.**Statement 3:** If ( m parallel n ), ( n perp beta ), and ( m parallel alpha ), then ( alpha perp beta ).Alright, so ( m ) is parallel to ( n ), and ( n ) is perpendicular to plane ( beta ). So, if ( n perp beta ), then ( n ) is a line that's perpendicular to every line in ( beta ) that it intersects. Since ( m parallel n ), then ( m ) must also be perpendicular to ( beta ), right? Because if two lines are parallel, they have the same direction, so if one is perpendicular to a plane, the other must be too.Now, ( m parallel alpha ). So, ( m ) is parallel to plane ( alpha ). If ( m ) is perpendicular to ( beta ) and parallel to ( alpha ), does that mean ( alpha ) is perpendicular to ( beta )?Hmm, let's think. If a line is parallel to a plane, it means the line is either lying on the plane or is parallel to some line on the plane. If that line is also perpendicular to another plane ( beta ), then the plane ( alpha ) must be perpendicular to ( beta ). Because if a line in ( alpha ) is perpendicular to ( beta ), then ( alpha ) and ( beta ) must intersect at a right angle.Wait, is that always true? Let me recall. If a line is parallel to a plane and perpendicular to another plane, then the two planes must be perpendicular. Yes, that seems correct.So, Statement 3 is correct.**Statement 4:** If ( m cap n = A ), ( m parallel alpha ), ( m parallel beta ), ( n parallel alpha ), and ( n parallel beta ), then ( alpha parallel beta ).Okay, so lines ( m ) and ( n ) intersect at point ( A ). Both ( m ) and ( n ) are parallel to both planes ( alpha ) and ( beta ). Does that mean ( alpha ) and ( beta ) are parallel?Hmm, if two intersecting lines are both parallel to two different planes, does that force the planes to be parallel?Let me visualize. If ( m ) and ( n ) intersect at ( A ), and both are parallel to ( alpha ) and ( beta ), then the direction of ( m ) and ( n ) must be parallel to both planes. Since ( m ) and ( n ) intersect, they define a plane themselves, say plane ( gamma ). If both ( alpha ) and ( beta ) are parallel to ( gamma ), then ( alpha ) and ( beta ) must be parallel to each other.Wait, is that right? If two planes are both parallel to the same plane ( gamma ), then they must be parallel to each other. Yes, that makes sense.Alternatively, since both ( m ) and ( n ) are parallel to ( alpha ) and ( beta ), and ( m ) and ( n ) are not parallel (they intersect), then the planes ( alpha ) and ( beta ) must be parallel because they both contain directions parallel to ( m ) and ( n ), which are not parallel themselves.Yes, so Statement 4 is correct.**Summary:**- Statement 1: Correct- Statement 2: Incorrect- Statement 3: Correct- Statement 4: CorrectSo, out of four statements, three are correct. Therefore, the answer should be C: 3.Wait, but hold on. Earlier, I thought Statement 1 was correct, but the initial assistant thought process said it was incorrect. Let me double-check Statement 1.**Rechecking Statement 1:** If ( m perp alpha ) and ( n parallel alpha ), then ( m perp n ).So, ( m ) is perpendicular to ( alpha ), meaning it's perpendicular to every line in ( alpha ). ( n ) is parallel to ( alpha ), so it's parallel to some line in ( alpha ). Therefore, ( m ) should be perpendicular to ( n ) because ( n ) is parallel to a line that ( m ) is perpendicular to.Yes, that seems correct. So, Statement 1 is correct.But the initial assistant thought process said Statement 1 was incorrect. Maybe I was confused earlier, but upon rethinking, I believe Statement 1 is correct.So, with that, all four statements except Statement 2 are correct. Wait, no, Statement 2 is incorrect, so Statements 1, 3, and 4 are correct. That's three correct statements.But hold on, in the initial problem, the options are A:1, B:2, C:3, D:4. So, if three are correct, the answer is C.But the initial assistant thought process concluded that Statements 2, 3, and 4 were correct, which would also be three. But in my analysis, Statements 1, 3, and 4 are correct, which is also three. So, regardless, the number of correct statements is three.Wait, but in the initial assistant thought process, Statement 1 was considered incorrect, but upon my analysis, it's correct. So, depending on that, the correct count could be three either way.But to be precise, let me confirm Statement 1 again.If ( m perp alpha ), then ( m ) is perpendicular to every line in ( alpha ). If ( n parallel alpha ), then ( n ) is parallel to some line in ( alpha ). Therefore, ( m ) must be perpendicular to ( n ). So, yes, Statement 1 is correct.Therefore, Statements 1, 3, and 4 are correct, making three correct statements. So, the answer is C:3.But wait, in the initial problem statement, the user provided a thought process where the assistant concluded that Statements 2, 3, and 4 were correct, which is also three. So, regardless of whether Statement 1 is correct or not, the number of correct statements is three.Wait, no. If Statement 1 is correct, then Statements 1, 3, and 4 are correct. If Statement 1 is incorrect, then Statements 2, 3, and 4 are correct. Either way, three statements are correct. So, the answer is C:3.But hold on, in my analysis, Statement 1 is correct, but the initial assistant thought process said it was incorrect. So, perhaps I need to reconcile that.Let me think again about Statement 1. If ( m perp alpha ) and ( n parallel alpha ), then ( m perp n ).Is this always true? Let's consider an example. Let ( alpha ) be the xy-plane. Then, ( m ) is a vertical line (along the z-axis). ( n ) is a line parallel to the xy-plane, say along the x-axis. Then, ( m ) is perpendicular to ( n ). So, in this case, it's true.But what if ( n ) is not in the xy-plane but parallel to it? For example, ( n ) could be a line parallel to the x-axis but shifted along the z-axis. Then, ( m ) is still along the z-axis, and ( n ) is along the x-axis direction. So, ( m ) is still perpendicular to ( n ).Wait, but in 3D space, two lines can be skew. So, if ( m ) is along the z-axis and ( n ) is along the x-axis shifted along y, are they still perpendicular?Wait, no. If ( n ) is parallel to the x-axis but shifted in y, then ( m ) is along z, and ( n ) is along x. The direction vectors are perpendicular, so the lines are perpendicular, even if they're skew.Wait, but in 3D, skew lines can have direction vectors that are perpendicular, but the lines themselves aren't necessarily perpendicular in the sense of intersecting at a right angle. So, does the statement mean that the lines are perpendicular in direction or that they intersect at a right angle?I think in geometry, when we say two lines are perpendicular, it usually means their direction vectors are perpendicular, regardless of whether they intersect. So, in that case, Statement 1 is correct because ( m ) is perpendicular to ( alpha ), so its direction is perpendicular to every direction in ( alpha ), and ( n ) is parallel to ( alpha ), so its direction is in ( alpha ), hence perpendicular to ( m )'s direction.Therefore, Statement 1 is correct.So, in conclusion, Statements 1, 3, and 4 are correct, making three correct statements. Therefore, the answer is C:3.

💡To find the smallest possible difference between a three-digit number ( a ) and a two-digit number ( b ) using the digits ( 1, 3, 5, 7, 8 ) exactly once each, follow these steps:1. **Minimize ( a ):** - The smallest three-digit number is formed by placing the smallest digits in the hundreds, tens, and units places. - The smallest digits available are ( 1, 3, ) and ( 5 ). - Therefore, ( a = 135 ).2. **Maximize ( b ):** - The largest two-digit number is formed by placing the largest remaining digits in the tens and units places. - The largest remaining digits are ( 7 ) and ( 8 ). - Therefore, ( b = 87 ).3. **Calculate the difference ( a - b ):** [ a - b = 135 - 87 = 48 ]The smallest possible difference is ( 48 ).

💡Okay, so I have this problem about a circle P on the coordinate plane. The center of the circle is at (1,2), and one endpoint of a diameter is at (4,6). I need to find the coordinates of the other endpoint of this diameter. Hmm, let me think about how to approach this.First, I remember that in a circle, the diameter is a straight line that passes through the center and has its endpoints on the circle. So, if I have one endpoint and the center, I should be able to find the other endpoint by using some properties of circles and coordinates.I think the key here is that the center of the circle is the midpoint of the diameter. That makes sense because a diameter is just a straight line passing through the center, so the center should be exactly halfway between the two endpoints. So, if I can find the midpoint between the two endpoints, that should give me the center of the circle.Wait, but I already know the center is (1,2), and one endpoint is (4,6). So, maybe I can use the midpoint formula to find the other endpoint. The midpoint formula is something like the average of the x-coordinates and the average of the y-coordinates of the two endpoints. Let me write that down.If I let the other endpoint be (x, y), then the midpoint between (4,6) and (x,y) should be (1,2). So, using the midpoint formula:Midpoint = ((4 + x)/2, (6 + y)/2) = (1,2)So, setting up the equations:(4 + x)/2 = 1and(6 + y)/2 = 2Now, I can solve these equations for x and y.Starting with the x-coordinate:(4 + x)/2 = 1Multiply both sides by 2:4 + x = 2Subtract 4 from both sides:x = 2 - 4x = -2Okay, so the x-coordinate of the other endpoint is -2.Now, let's solve for the y-coordinate:(6 + y)/2 = 2Multiply both sides by 2:6 + y = 4Subtract 6 from both sides:y = 4 - 6y = -2So, the y-coordinate of the other endpoint is -2.Putting it all together, the other endpoint of the diameter is at (-2, -2).Wait, let me double-check that. If the midpoint between (4,6) and (-2,-2) is indeed (1,2), then I can verify it.Calculating the midpoint:x-coordinate: (4 + (-2))/2 = (2)/2 = 1y-coordinate: (6 + (-2))/2 = (4)/2 = 2Yes, that gives the midpoint as (1,2), which matches the center of the circle. So, that seems correct.Alternatively, I could think about the vector from the center to the known endpoint and then go the same distance in the opposite direction to find the other endpoint.From the center (1,2) to the endpoint (4,6), the change in x is 4 - 1 = 3, and the change in y is 6 - 2 = 4. So, the vector from the center to (4,6) is (3,4). To find the other endpoint, I need to go in the opposite direction from the center, which would be subtracting this vector from the center.So, starting at the center (1,2), subtracting the vector (3,4):x-coordinate: 1 - 3 = -2y-coordinate: 2 - 4 = -2Again, that gives me (-2, -2). So, both methods give the same result, which makes me more confident that it's correct.I think I've got it. The other endpoint of the diameter is at (-2, -2).

💡Okay, so I have this problem with two parts. Let me try to tackle them one by one. First, part 1: The line l intersects the curve C₁ at points A and B. I need to find the length |AB|. Alright, let's see. The line l is given parametrically as:[begin{cases}x = 1 + frac{1}{2}t y = frac{sqrt{3}}{2}tend{cases}]where t is a parameter. And the curve C₁ is given parametrically as:[begin{cases}x = costheta y = sinthetaend{cases}]where θ is a parameter. Hmm, so curve C₁ looks like the unit circle because x² + y² = cos²θ + sin²θ = 1. So, it's a circle with radius 1 centered at the origin.Line l is a straight line, and I need to find where it intersects the unit circle. To find the points of intersection, I can substitute the parametric equations of the line into the equation of the circle.So, substituting x and y from line l into x² + y² = 1:[left(1 + frac{1}{2}tright)^2 + left(frac{sqrt{3}}{2}tright)^2 = 1]Let me expand this:First, expand (left(1 + frac{1}{2}tright)^2):[1 + t + frac{1}{4}t^2]Then, expand (left(frac{sqrt{3}}{2}tright)^2):[frac{3}{4}t^2]So, adding them together:[1 + t + frac{1}{4}t^2 + frac{3}{4}t^2 = 1]Combine like terms:[1 + t + left(frac{1}{4} + frac{3}{4}right)t^2 = 1][1 + t + t^2 = 1]Subtract 1 from both sides:[t + t^2 = 0][t(t + 1) = 0]So, t = 0 or t = -1.Now, plug these t-values back into the parametric equations of line l to find points A and B.For t = 0:[x = 1 + frac{1}{2}(0) = 1 y = frac{sqrt{3}}{2}(0) = 0]So, point A is (1, 0).For t = -1:[x = 1 + frac{1}{2}(-1) = 1 - frac{1}{2} = frac{1}{2} y = frac{sqrt{3}}{2}(-1) = -frac{sqrt{3}}{2}]So, point B is (left(frac{1}{2}, -frac{sqrt{3}}{2}right)).Now, to find the distance |AB|, I can use the distance formula:[|AB| = sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}]Plugging in the coordinates:[|AB| = sqrt{left(frac{1}{2} - 1right)^2 + left(-frac{sqrt{3}}{2} - 0right)^2}]Simplify:[|AB| = sqrt{left(-frac{1}{2}right)^2 + left(-frac{sqrt{3}}{2}right)^2}][|AB| = sqrt{frac{1}{4} + frac{3}{4}} = sqrt{1} = 1]So, the length |AB| is 1.Alright, that seems straightforward. Now, moving on to part 2.Part 2: Curve C₂ is given as:[begin{cases}x = frac{1}{2}costheta y = frac{sqrt{3}}{2}sinthetaend{cases}]where θ is a parameter. Point P is a moving point on curve C₂. I need to find the minimum distance from point P to line l.First, let me understand curve C₂. It looks like an ellipse because x and y are scaled versions of cosine and sine. Specifically, it's an ellipse with semi-major axis (frac{sqrt{3}}{2}) along the y-axis and semi-minor axis (frac{1}{2}) along the x-axis.So, parametric equations for an ellipse are usually:[x = acostheta y = bsintheta]where a and b are the semi-axes. In this case, a = 1/2 and b = (sqrt{3}/2).Now, I need to find the minimum distance from any point P on this ellipse to the line l.First, let me recall the formula for the distance from a point (x₀, y₀) to a line ax + by + c = 0:[d = frac{|ax₀ + by₀ + c|}{sqrt{a^2 + b^2}}]So, I need to write line l in the standard form ax + by + c = 0.From the parametric equations of line l:[x = 1 + frac{1}{2}t y = frac{sqrt{3}}{2}t]I can eliminate the parameter t to get the equation of the line.From the second equation, solve for t:[y = frac{sqrt{3}}{2}t implies t = frac{2y}{sqrt{3}}]Plug this into the first equation:[x = 1 + frac{1}{2}left(frac{2y}{sqrt{3}}right) = 1 + frac{y}{sqrt{3}}]So, rearranged:[x - 1 = frac{y}{sqrt{3}} implies sqrt{3}(x - 1) = y]Bring all terms to one side:[sqrt{3}x - y - sqrt{3} = 0]So, the standard form is:[sqrt{3}x - y - sqrt{3} = 0]Thus, a = (sqrt{3}), b = -1, c = -(sqrt{3}).Now, point P on curve C₂ is (left(frac{1}{2}costheta, frac{sqrt{3}}{2}sinthetaright)). Let me denote this as (x, y).So, the distance d from P to line l is:[d = frac{|sqrt{3}x - y - sqrt{3}|}{sqrt{(sqrt{3})^2 + (-1)^2}} = frac{|sqrt{3}x - y - sqrt{3}|}{sqrt{3 + 1}} = frac{|sqrt{3}x - y - sqrt{3}|}{2}]Now, substitute x and y with the parametric equations of C₂:[x = frac{1}{2}costheta y = frac{sqrt{3}}{2}sintheta]So, plug these into the distance formula:[d = frac{|sqrt{3}left(frac{1}{2}costhetaright) - left(frac{sqrt{3}}{2}sinthetaright) - sqrt{3}|}{2}]Simplify step by step.First, compute each term:[sqrt{3}left(frac{1}{2}costhetaright) = frac{sqrt{3}}{2}costheta][- left(frac{sqrt{3}}{2}sinthetaright) = -frac{sqrt{3}}{2}sintheta][- sqrt{3} = -sqrt{3}]So, combining these:[frac{sqrt{3}}{2}costheta - frac{sqrt{3}}{2}sintheta - sqrt{3}]Factor out (frac{sqrt{3}}{2}) from the first two terms:[frac{sqrt{3}}{2}(costheta - sintheta) - sqrt{3}]Now, let me write the entire expression inside the absolute value:[left|frac{sqrt{3}}{2}(costheta - sintheta) - sqrt{3}right|]Factor out (sqrt{3}):[sqrt{3}left|frac{1}{2}(costheta - sintheta) - 1right|]So, the distance becomes:[d = frac{sqrt{3}left|frac{1}{2}(costheta - sintheta) - 1right|}{2} = frac{sqrt{3}}{2}left|frac{1}{2}(costheta - sintheta) - 1right|]Simplify further:[d = frac{sqrt{3}}{2}left|frac{costheta - sintheta}{2} - 1right| = frac{sqrt{3}}{2}left|frac{costheta - sintheta - 2}{2}right| = frac{sqrt{3}}{4}|costheta - sintheta - 2|]Wait, that seems a bit messy. Maybe I made a miscalculation.Let me go back a step. After factoring out (sqrt{3}), we had:[sqrt{3}left|frac{1}{2}(costheta - sintheta) - 1right|]So, the distance is:[d = frac{sqrt{3}}{2} times left|frac{1}{2}(costheta - sintheta) - 1right|]Which is:[d = frac{sqrt{3}}{2} times left|frac{costheta - sintheta}{2} - 1right| = frac{sqrt{3}}{2} times left|frac{costheta - sintheta - 2}{2}right|]Which simplifies to:[d = frac{sqrt{3}}{4}|costheta - sintheta - 2|]Hmm, that seems correct. But maybe I can express (costheta - sintheta) in a different form to make it easier to find the minimum.Recall that expressions like (acostheta + bsintheta) can be written as (Rcos(theta + phi)) where (R = sqrt{a^2 + b^2}) and (phi = arctanleft(frac{b}{a}right)) or something like that.In this case, we have (costheta - sintheta). Let me write this as:[costheta - sintheta = sqrt{2}left(frac{1}{sqrt{2}}costheta - frac{1}{sqrt{2}}sinthetaright)]Which is:[sqrt{2}cosleft(theta + frac{pi}{4}right)]Because (costhetacosphi - sinthetasinphi = cos(theta + phi)), and here (phi = frac{pi}{4}) since (cosphi = sinphi = frac{1}{sqrt{2}}).So, substituting back:[costheta - sintheta = sqrt{2}cosleft(theta + frac{pi}{4}right)]Therefore, the expression inside the absolute value becomes:[sqrt{2}cosleft(theta + frac{pi}{4}right) - 2]So, the distance d is:[d = frac{sqrt{3}}{4}|sqrt{2}cosleft(theta + frac{pi}{4}right) - 2|]Now, to find the minimum distance, I need to minimize d with respect to θ.Since the absolute value is always non-negative, the minimum occurs when the expression inside is minimized.Let me denote:[f(theta) = sqrt{2}cosleft(theta + frac{pi}{4}right) - 2]So, |f(θ)| is what we're dealing with.But since we have a negative sign, let me think about the range of f(θ).The cosine function ranges between -1 and 1, so:[sqrt{2}cosleft(theta + frac{pi}{4}right) in [-sqrt{2}, sqrt{2}]]Therefore:[f(theta) = sqrt{2}cosleft(theta + frac{pi}{4}right) - 2 in [-sqrt{2} - 2, sqrt{2} - 2]]Which is approximately [-3.414, -0.586].So, f(θ) is always negative because the maximum value is (sqrt{2} - 2 approx -0.586), which is still negative. Therefore, |f(θ)| = |negative| = positive, so:[|f(theta)| = | sqrt{2}cosleft(theta + frac{pi}{4}right) - 2 | = 2 - sqrt{2}cosleft(theta + frac{pi}{4}right)]Because when the inside is negative, the absolute value flips the sign.So, now, the distance d becomes:[d = frac{sqrt{3}}{4} left(2 - sqrt{2}cosleft(theta + frac{pi}{4}right)right)]Now, to minimize d, we need to minimize the expression inside the parentheses, which is:[2 - sqrt{2}cosleft(theta + frac{pi}{4}right)]Since we are subtracting (sqrt{2}cos(cdot)), the expression is minimized when (cosleft(theta + frac{pi}{4}right)) is maximized because that would subtract the largest possible value, making the whole expression as small as possible.The maximum value of (cosleft(theta + frac{pi}{4}right)) is 1. So, the minimum value of the expression is:[2 - sqrt{2}(1) = 2 - sqrt{2}]Therefore, the minimum distance d is:[d_{text{min}} = frac{sqrt{3}}{4}(2 - sqrt{2}) = frac{sqrt{3}}{4} times (2 - sqrt{2})]Simplify:[d_{text{min}} = frac{2sqrt{3} - sqrt{6}}{4} = frac{sqrt{3}}{2} - frac{sqrt{6}}{4}]But let me check if this is correct. Wait, actually, I think I made a mistake in the sign when considering the absolute value.Earlier, I concluded that f(θ) is always negative, so |f(θ)| = -f(θ) = 2 - sqrt{2}cos(theta + pi/4). Therefore, to minimize d, which is proportional to |f(θ)|, we need to minimize |f(θ)|, which is equivalent to minimizing 2 - sqrt{2}cos(theta + pi/4).But since 2 - sqrt{2}cos(theta + pi/4) is minimized when (cos(theta + pi/4)) is maximized, which is 1. So, the minimum value is 2 - sqrt{2}, and thus the minimum distance is (frac{sqrt{3}}{4}(2 - sqrt{2})).Wait, but 2 - sqrt{2} is approximately 0.5858, which is positive, so the minimum distance is positive, which makes sense.But let me double-check if this is indeed the minimum. Alternatively, maybe I should consider the expression differently.Alternatively, perhaps I can express the distance as a function of θ and then take the derivative to find the minimum.Let me try that approach as a verification.So, the distance squared is often easier to work with because the square root can complicate differentiation. But since we're dealing with absolute values, maybe it's manageable.But let's see.We have:[d = frac{sqrt{3}}{4}|sqrt{2}cosleft(theta + frac{pi}{4}right) - 2|]As we established, this is equal to:[d = frac{sqrt{3}}{4}(2 - sqrt{2}cosleft(theta + frac{pi}{4}right))]So, to minimize d, we need to minimize the expression (2 - sqrt{2}cosleft(theta + frac{pi}{4}right)).Since (cosleft(theta + frac{pi}{4}right)) has a maximum of 1 and a minimum of -1, the expression (2 - sqrt{2}cos(cdot)) will have its minimum when (cos(cdot)) is maximum, i.e., 1, giving (2 - sqrt{2}), and its maximum when (cos(cdot)) is minimum, i.e., -1, giving (2 + sqrt{2}).Therefore, the minimum distance is indeed:[d_{text{min}} = frac{sqrt{3}}{4}(2 - sqrt{2}) = frac{sqrt{3}}{4} times 2 - frac{sqrt{3}}{4} times sqrt{2} = frac{sqrt{3}}{2} - frac{sqrt{6}}{4}]But let me rationalize or simplify this expression further if possible.Alternatively, factor out (frac{sqrt{3}}{4}):[d_{text{min}} = frac{sqrt{3}}{4}(2 - sqrt{2}) = frac{sqrt{3}(2 - sqrt{2})}{4}]Which can also be written as:[d_{text{min}} = frac{2sqrt{3} - sqrt{6}}{4}]But perhaps the problem expects the answer in a different form. Let me see if I can express it differently.Alternatively, factor out (sqrt{2}) from the numerator:[d_{text{min}} = frac{sqrt{3}}{4} times (2 - sqrt{2}) = frac{sqrt{3}}{4} times sqrt{2} times left(sqrt{2} - 1right)]Wait, because (2 - sqrt{2} = sqrt{2}(sqrt{2} - 1)). Let me check:[sqrt{2}(sqrt{2} - 1) = 2 - sqrt{2}]Yes, that's correct. So, substituting back:[d_{text{min}} = frac{sqrt{3}}{4} times sqrt{2} times (sqrt{2} - 1) = frac{sqrt{6}}{4} (sqrt{2} - 1)]Because (sqrt{3} times sqrt{2} = sqrt{6}).So, the minimum distance is:[d_{text{min}} = frac{sqrt{6}}{4} (sqrt{2} - 1)]That seems like a more compact form. Let me compute this numerically to check if it makes sense.Compute (sqrt{2} approx 1.4142), so (sqrt{2} - 1 approx 0.4142).Compute (sqrt{6} approx 2.4495).So, (frac{sqrt{6}}{4} (sqrt{2} - 1) approx frac{2.4495}{4} times 0.4142 approx 0.6124 times 0.4142 approx 0.2535).Earlier, we had (d_{text{min}} = frac{sqrt{3}}{4}(2 - sqrt{2}) approx frac{1.732}{4} times (2 - 1.4142) approx 0.433 times 0.5858 approx 0.2535). So, consistent.Therefore, the minimum distance is (frac{sqrt{6}}{4} (sqrt{2} - 1)).But let me check if this is indeed the minimum. Alternatively, perhaps I should consider the expression without factoring.Wait, another approach: since the distance is given by:[d = frac{sqrt{3}}{4}|sqrt{2}cos(theta + pi/4) - 2|]We can think of this as a function of θ, and to find its minimum, we can consider the range of the cosine function.Since (cos(theta + pi/4)) ranges between -1 and 1, the expression inside the absolute value, (sqrt{2}cos(theta + pi/4) - 2), ranges between (-sqrt{2} - 2) and (sqrt{2} - 2).As we saw earlier, the maximum value of the expression inside the absolute value is (sqrt{2} - 2 approx -0.5858), which is negative, so the absolute value makes it positive, and the minimum distance occurs when this expression is closest to zero, i.e., when (sqrt{2}cos(theta + pi/4) - 2) is closest to zero from below.Wait, actually, the minimum distance occurs when the expression inside the absolute value is minimized in magnitude. Since the expression is always negative, the minimum |expression| occurs when the expression is closest to zero, i.e., when (sqrt{2}cos(theta + pi/4)) is as close to 2 as possible. But since (sqrt{2}cos(theta + pi/4)) can only go up to (sqrt{2} approx 1.4142), which is less than 2, the closest it can get is when (cos(theta + pi/4) = 1), making the expression (sqrt{2} - 2), which is the maximum negative value, but in absolute terms, it's the smallest positive value.Wait, that seems contradictory. Let me clarify.The expression inside the absolute value is (sqrt{2}cos(theta + pi/4) - 2). Since (sqrt{2}cos(theta + pi/4)) can only go up to (sqrt{2}), the expression can only go up to (sqrt{2} - 2), which is approximately -0.5858. So, the expression is always negative, and the absolute value is always positive.Therefore, the minimum distance occurs when the expression inside the absolute value is closest to zero, i.e., when (sqrt{2}cos(theta + pi/4)) is as large as possible, which is (sqrt{2}). Therefore, the minimum |expression| is (2 - sqrt{2}), which is approximately 0.5858.Wait, but that would mean the minimum distance is (frac{sqrt{3}}{4}(2 - sqrt{2})), which is approximately 0.2535, as before.But let me think again: since the expression inside the absolute value is (sqrt{2}cos(theta + pi/4) - 2), which is always negative, the absolute value is (2 - sqrt{2}cos(theta + pi/4)). To minimize this, we need to maximize (sqrt{2}cos(theta + pi/4)), which is achieved when (cos(theta + pi/4) = 1), giving the minimum value of (2 - sqrt{2}).Therefore, the minimum distance is indeed (frac{sqrt{3}}{4}(2 - sqrt{2})), which simplifies to (frac{sqrt{6}}{4}(sqrt{2} - 1)).Wait, but (sqrt{6}/4 (sqrt{2} - 1)) is the same as (sqrt{3}/4 (2 - sqrt{2})), because:[sqrt{6} = sqrt{2 times 3} = sqrt{2}sqrt{3}]So,[frac{sqrt{6}}{4} (sqrt{2} - 1) = frac{sqrt{2}sqrt{3}}{4} (sqrt{2} - 1) = frac{sqrt{3}}{4} (sqrt{2}(sqrt{2} - 1)) = frac{sqrt{3}}{4} (2 - sqrt{2})]Yes, that's correct.So, both forms are equivalent. The problem might expect the answer in the form with (sqrt{6}), so I'll go with that.Therefore, the minimum distance is (frac{sqrt{6}}{4}(sqrt{2} - 1)).But let me check if this makes sense geometrically. Curve C₂ is an ellipse inside the unit circle, and line l is a straight line intersecting the unit circle at points A and B. The minimum distance from the ellipse to the line should be positive and less than the distance from the origin to the line.Wait, what's the distance from the origin to line l?Using the distance formula:[d = frac{|sqrt{3}(0) - 1(0) - sqrt{3}|}{sqrt{(sqrt{3})^2 + (-1)^2}} = frac{|,-sqrt{3}|}{2} = frac{sqrt{3}}{2} approx 0.866]So, the minimum distance from the ellipse to the line is approximately 0.2535, which is less than (sqrt{3}/2), which makes sense because the ellipse is closer to the line than the origin.Wait, but actually, the origin is inside the ellipse? No, the ellipse is centered at the origin with semi-axes 1/2 and (sqrt{3}/2), so it's entirely within the unit circle. The line l passes through (1,0) and (1/2, -√3/2), which are on the unit circle. So, the ellipse is inside the circle, and the line passes through the circle, so the minimum distance from the ellipse to the line should be the shortest distance from any point on the ellipse to the line.Given that the ellipse is closer to the line than the origin, the minimum distance should be less than the distance from the origin to the line, which is (sqrt{3}/2 approx 0.866). Our calculated minimum distance is approximately 0.2535, which is indeed less, so that makes sense.Alternatively, maybe I can visualize this. The line l has a slope of (sqrt{3}), which is 60 degrees from the x-axis. The ellipse is stretched more along the y-axis. The closest point on the ellipse to the line would likely be in the lower half of the ellipse, given the line's slope.But regardless, the calculations seem consistent.So, to summarize:1. The length |AB| is 1.2. The minimum distance from point P on curve C₂ to line l is (frac{sqrt{6}}{4}(sqrt{2} - 1)).I think that's it. I've double-checked the steps, and everything seems consistent.

💡To determine the value of ( x ) such that the point ( (x, -6) ) lies on the straight line joining the points ( (0, 10) ) and ( (-8, 0) ), we can follow these steps:1. **Calculate the slope of the line joining ( (0, 10) ) and ( (-8, 0) ):** The slope ( m ) between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is given by: [ m = frac{y_2 - y_1}{x_2 - x_1} ] Substituting the given points: [ m = frac{0 - 10}{-8 - 0} = frac{-10}{-8} = frac{5}{4} ]2. **Set up the equation for the slope between ( (0, 10) ) and ( (x, -6) ):** Since the point ( (x, -6) ) lies on the same line, the slope between ( (0, 10) ) and ( (x, -6) ) must also be ( frac{5}{4} ). Therefore: [ frac{-6 - 10}{x - 0} = frac{5}{4} ] Simplifying the numerator: [ frac{-16}{x} = frac{5}{4} ]3. **Solve for ( x ):** Cross-multiplying to solve for ( x ): [ -16 cdot 4 = 5x implies -64 = 5x implies x = frac{-64}{5} ]Thus, the value of ( x ) is ( -frac{64}{5} ).[boxed{textbf{(D) } -frac{64}{5}}]

💡First, I need to determine the weight of walnuts in the mixture by subtracting the weight of almonds from the total weight of the mixture.Next, I'll set up a proportion to find out how many parts walnuts correspond to the given weight of walnuts, based on the known ratio of almonds.Finally, I'll solve for the unknown variable to establish the ratio of almonds to walnuts in the mixture.