answer：First, I need to calculate the volume of each cuboid using the formula for the volume of a cuboid, which is length multiplied by width multiplied by height.For the first cuboid with a height of 7 cm:- Length = 44 cm- Width = 35 cm- Height = 7 cm- Volume = 44 cm × 35 cm × 7 cm = 10780 cm³For the second cuboid with a height of 3 cm:- Length = 44 cm- Width = 35 cm- Height = 3 cm- Volume = 44 cm × 35 cm × 3 cm = 4620 cm³Finally, I add the volumes of the two cuboids to find the total volume:- Total Volume = 10780 cm³ + 4620 cm³ = 15400 cm³

answer：Okay, so I have this triangle ABC, and the sides opposite angles A, B, and C are a, b, and c respectively. They've given me the lengths: a is 3, b is 4, and c is 6. I need to find cos B. Hmm, okay, let me think about how to approach this.First, I remember that in a triangle, if I know all three sides, I can use the Law of Cosines to find any of the angles. The Law of Cosines formula is something like c² = a² + b² - 2ab cos C, right? So, if I need to find cos B, I should rearrange the formula to solve for cos B.Let me write down the formula for angle B. Since angle B is opposite side b, which is 4, the sides adjacent to angle B are a and c, which are 3 and 6. So, plugging into the Law of Cosines formula, it should be:b² = a² + c² - 2ac cos BWait, is that right? Let me double-check. Yes, because in the Law of Cosines, the side opposite the angle is squared on the left, and the other two sides are squared on the right, subtracted by twice the product of those sides and the cosine of the angle. So, yes, that formula makes sense.So, plugging in the values I have:4² = 3² + 6² - 2 * 3 * 6 * cos BCalculating each term step by step. 4 squared is 16, 3 squared is 9, and 6 squared is 36. So, substituting those in:16 = 9 + 36 - 2 * 3 * 6 * cos BLet me compute 9 + 36 first. That's 45. So now the equation is:16 = 45 - 2 * 3 * 6 * cos BNext, compute 2 * 3 * 6. 2 times 3 is 6, times 6 is 36. So, that simplifies to:16 = 45 - 36 cos BNow, I need to solve for cos B. Let me rearrange the equation. Subtract 45 from both sides:16 - 45 = -36 cos B16 minus 45 is -29. So:-29 = -36 cos BNow, divide both sides by -36 to solve for cos B:cos B = (-29) / (-36)The negatives cancel out, so cos B = 29/36.Wait, let me check my steps again to make sure I didn't make a mistake. Starting from the Law of Cosines:b² = a² + c² - 2ac cos BPlugging in the numbers:16 = 9 + 36 - 36 cos BYes, that's correct. Then 9 + 36 is 45, so 16 = 45 - 36 cos B. Subtract 45: 16 - 45 is -29, so -29 = -36 cos B. Dividing both sides by -36 gives cos B = 29/36.Looking at the answer choices, option C is 29/36, so that must be the correct answer.I think that's solid. I don't see any mistakes in my calculations, so I feel confident with this answer.**Final Answer**boxed{C}

answer：First, I need to compute ( g(4) ). Since ( 4 geq 3 ), I use the first case of the piecewise function:[g(4) = -4^2 + 1 = -16 + 1 = -15]Next, I compute ( g(-15) ). Since ( -15 < 3 ), I use the second case:[g(-15) = -15 + 10 = -5]Then, I compute ( g(-5) ). Again, ( -5 < 3 ), so I use the second case:[g(-5) = -5 + 10 = 5]Now, I compute ( g(5) ). Since ( 5 geq 3 ), I use the first case:[g(5) = -5^2 + 1 = -25 + 1 = -24]Finally, I compute ( g(-24) ). Since ( -24 < 3 ), I use the second case:[g(-24) = -24 + 10 = -14]Therefore, the value of ( g(g(g(g(g(4))))) ) is (-14).

answer：Okay, so I have this problem where a line intersects a parabola at two points, A and B. The line is given by the equation y = kx - k, and the parabola is y² = 4x. The distance between points A and B is 4, and I need to find the distance from the midpoint of chord AB to the y-axis. The options are A: 3/4, B: 1, C: 2, D: 4/3.First, I should visualize what's going on here. The parabola y² = 4x is a standard right-opening parabola with vertex at the origin. Its focus is at (1, 0), and the directrix is the line x = -1. The line given is y = kx - k, which can be rewritten as y = k(x - 1). So this line passes through the point (1, 0) regardless of the value of k. That's interesting because (1, 0) is the focus of the parabola.Since the line passes through the focus, it's a focal chord of the parabola. I remember that for a parabola, the focal chord has some special properties. Maybe that can help me here.I need to find the points of intersection between the line and the parabola. So let's substitute y from the line equation into the parabola equation.Starting with the parabola equation:y² = 4xSubstitute y = k(x - 1):[k(x - 1)]² = 4xExpanding the left side:k²(x - 1)² = 4xk²(x² - 2x + 1) = 4xNow, distribute k²:k²x² - 2k²x + k² = 4xBring all terms to one side to form a quadratic equation:k²x² - (2k² + 4)x + k² = 0So, quadratic in x: ax² + bx + c = 0, where:a = k²b = -(2k² + 4)c = k²Let me denote this as:k²x² - (2k² + 4)x + k² = 0Now, let's find the solutions for x. The quadratic formula is x = [-b ± sqrt(b² - 4ac)] / (2a)Plugging in the values:x = [2k² + 4 ± sqrt{(2k² + 4)² - 4 * k² * k²}] / (2k²)Simplify the discriminant:(2k² + 4)² - 4k⁴= 4k⁴ + 16k² + 16 - 4k⁴= 16k² + 16= 16(k² + 1)So, sqrt of discriminant is 4*sqrt(k² + 1)Therefore, x = [2k² + 4 ± 4sqrt(k² + 1)] / (2k²)Simplify numerator and denominator:Factor out 2 in numerator:x = [2(k² + 2) ± 4sqrt(k² + 1)] / (2k²)= [ (k² + 2) ± 2sqrt(k² + 1) ] / k²So, the two x-coordinates of points A and B are:x₁ = [k² + 2 + 2sqrt(k² + 1)] / k²x₂ = [k² + 2 - 2sqrt(k² + 1)] / k²Now, let's find the corresponding y-coordinates using y = k(x - 1)For x₁:y₁ = k( [k² + 2 + 2sqrt(k² + 1)] / k² - 1 )= k( [k² + 2 + 2sqrt(k² + 1) - k²] / k² )= k( [2 + 2sqrt(k² + 1)] / k² )= [2k + 2k sqrt(k² + 1)] / k²= [2(1 + sqrt(k² + 1))] / kSimilarly, for x₂:y₂ = k( [k² + 2 - 2sqrt(k² + 1)] / k² - 1 )= k( [k² + 2 - 2sqrt(k² + 1) - k²] / k² )= k( [2 - 2sqrt(k² + 1)] / k² )= [2k - 2k sqrt(k² + 1)] / k²= [2(1 - sqrt(k² + 1))] / kSo, points A and B are:A: ( [k² + 2 + 2sqrt(k² + 1)] / k² , [2(1 + sqrt(k² + 1))] / k )B: ( [k² + 2 - 2sqrt(k² + 1)] / k² , [2(1 - sqrt(k² + 1))] / k )Now, I need to find the distance between A and B, which is given as 4.Let me compute the distance AB.Distance formula: sqrt[(x₁ - x₂)² + (y₁ - y₂)²]Compute x₁ - x₂:x₁ - x₂ = [k² + 2 + 2sqrt(k² + 1)] / k² - [k² + 2 - 2sqrt(k² + 1)] / k²= [4sqrt(k² + 1)] / k²Similarly, y₁ - y₂:y₁ - y₂ = [2(1 + sqrt(k² + 1))] / k - [2(1 - sqrt(k² + 1))] / k= [4 sqrt(k² + 1)] / kSo, distance AB:sqrt[ (4sqrt(k² + 1)/k²)² + (4 sqrt(k² + 1)/k)² ]Compute each term inside sqrt:First term: (4sqrt(k² + 1)/k²)² = 16(k² + 1)/k⁴Second term: (4 sqrt(k² + 1)/k)² = 16(k² + 1)/k²So, total inside sqrt:16(k² + 1)/k⁴ + 16(k² + 1)/k²Factor out 16(k² + 1)/k⁴:= 16(k² + 1)/k⁴ [1 + k²]Wait, 16(k² + 1)/k⁴ + 16(k² + 1)/k² = 16(k² + 1)/k⁴ + 16(k² + 1)k²/k⁴= [16(k² + 1) + 16(k² + 1)k²] / k⁴= 16(k² + 1)(1 + k²) / k⁴= 16(k² + 1)² / k⁴Therefore, distance AB:sqrt[16(k² + 1)² / k⁴] = 4(k² + 1)/k²Given that distance AB is 4, so:4(k² + 1)/k² = 4Divide both sides by 4:(k² + 1)/k² = 1Multiply both sides by k²:k² + 1 = k²Subtract k²:1 = 0Wait, that can't be right. I must have made a mistake somewhere.Let me go back through my steps.Starting from the quadratic equation:k²x² - (2k² + 4)x + k² = 0Discriminant: (2k² + 4)^2 - 4*k²*k² = 4k^4 + 16k² + 16 - 4k^4 = 16k² + 16Which is correct.Then, sqrt discriminant is 4*sqrt(k² + 1). Correct.Then, x = [2k² + 4 ± 4sqrt(k² + 1)] / (2k²) = [k² + 2 ± 2sqrt(k² + 1)] / k². Correct.Then, y = k(x - 1). So, for x₁, y₁ = k([k² + 2 + 2sqrt(k² + 1)] / k² - 1) = k([k² + 2 + 2sqrt(k² + 1) - k²] / k²) = k([2 + 2sqrt(k² + 1)] / k²) = [2k + 2k sqrt(k² + 1)] / k² = [2(1 + sqrt(k² + 1))] / k. Correct.Similarly for y₂. Correct.Then, x₁ - x₂ = [4sqrt(k² + 1)] / k². Correct.y₁ - y₂ = [4 sqrt(k² + 1)] / k. Correct.Then, distance AB: sqrt[ (4sqrt(k² + 1)/k²)^2 + (4 sqrt(k² + 1)/k)^2 ]Compute first term: (4sqrt(k² + 1)/k²)^2 = 16(k² + 1)/k^4Second term: (4 sqrt(k² + 1)/k)^2 = 16(k² + 1)/k²So, adding them: 16(k² + 1)/k^4 + 16(k² + 1)/k²Factor out 16(k² + 1)/k^4:= 16(k² + 1)/k^4 [1 + k²]Wait, 16(k² + 1)/k^4 + 16(k² + 1)/k² = 16(k² + 1)/k^4 + 16(k² + 1)k²/k^4 = [16(k² + 1) + 16(k² + 1)k²]/k^4= 16(k² + 1)(1 + k²)/k^4= 16(k² + 1)^2 /k^4So, sqrt of that is 4(k² + 1)/k²Set equal to 4:4(k² + 1)/k² = 4 => (k² + 1)/k² = 1 => k² + 1 = k² => 1 = 0This is impossible, so I must have messed up somewhere.Wait, maybe I made a mistake in computing the distance AB.Let me compute AB again.Points A and B:A: (x₁, y₁) = ( [k² + 2 + 2sqrt(k² + 1)] / k² , [2(1 + sqrt(k² + 1))] / k )B: (x₂, y₂) = ( [k² + 2 - 2sqrt(k² + 1)] / k² , [2(1 - sqrt(k² + 1))] / k )So, x₁ - x₂ = [4sqrt(k² + 1)] / k²y₁ - y₂ = [4 sqrt(k² + 1)] / kSo, distance AB is sqrt[(4sqrt(k² + 1)/k²)^2 + (4 sqrt(k² + 1)/k)^2]Compute each term:(4sqrt(k² + 1)/k²)^2 = 16(k² + 1)/k^4(4 sqrt(k² + 1)/k)^2 = 16(k² + 1)/k²So, total inside sqrt:16(k² + 1)/k^4 + 16(k² + 1)/k²Factor out 16(k² + 1)/k^4:= 16(k² + 1)/k^4 [1 + k²]Wait, 16(k² + 1)/k^4 + 16(k² + 1)/k² = 16(k² + 1)(1/k^4 + 1/k²) = 16(k² + 1)(1 + k²)/k^4Yes, that's correct.So, sqrt[16(k² + 1)^2 /k^4] = 4(k² + 1)/k²Set equal to 4:4(k² + 1)/k² = 4Divide both sides by 4:(k² + 1)/k² = 1Multiply both sides by k²:k² + 1 = k²Subtract k²:1 = 0This is a contradiction, which suggests that my approach is wrong.Wait, maybe I made a mistake in calculating the distance AB.Alternatively, perhaps the line intersects the parabola at only one point, but the problem says it intersects at two points A and B.Alternatively, maybe I should approach this differently.Since the line passes through the focus (1,0), and the parabola is y² = 4x, which is a standard parabola with focus at (1,0).In a parabola, the focal chord has the property that the product of the distances from the focus to each end of the chord is equal to the square of the focal length.Wait, but I'm not sure about that.Alternatively, maybe I can use parametric equations for the parabola.For parabola y² = 4x, parametric equations are x = t², y = 2t.So, any point on the parabola can be written as (t², 2t).Now, the line is y = kx - k.So, substitute y = 2t and x = t² into the line equation:2t = k t² - kRearrange:k t² - 2t - k = 0This is a quadratic in t: k t² - 2t - k = 0Let me solve for t:t = [2 ± sqrt(4 + 4k²)] / (2k) = [2 ± 2sqrt(1 + k²)] / (2k) = [1 ± sqrt(1 + k²)] / kSo, the two parameters t₁ and t₂ are:t₁ = [1 + sqrt(1 + k²)] / kt₂ = [1 - sqrt(1 + k²)] / kSo, points A and B are:A: (t₁², 2t₁) = ( [1 + sqrt(1 + k²)]² / k² , 2[1 + sqrt(1 + k²)] / k )B: (t₂², 2t₂) = ( [1 - sqrt(1 + k²)]² / k² , 2[1 - sqrt(1 + k²)] / k )Wait, this seems similar to what I had before, but let's see.Compute the distance AB.Using parametric forms, the distance between A and B can be found using the distance formula.But perhaps it's easier to use the property of the parabola.In a parabola, the distance between two points with parameters t₁ and t₂ is given by sqrt[(t₁² - t₂²)^2 + (2t₁ - 2t₂)^2]But that might not be straightforward.Alternatively, since the line passes through the focus, maybe we can use the property that the length of the focal chord is 4a / sin²θ, where θ is the angle the chord makes with the x-axis.But in this case, the parabola is y² = 4x, so a = 1.So, length of focal chord is 4 / sin²θ.Given that the length is 4, so 4 = 4 / sin²θ => sin²θ = 1 => sinθ = ±1 => θ = 90° or 270°, which would mean the line is vertical.But the line is y = kx - k, which is vertical only if k is infinite, which is not the case here.Hmm, that seems contradictory.Alternatively, maybe I made a mistake in recalling the formula.Wait, the formula for the length of a focal chord in a parabola y² = 4ax is 4a / sin²θ, where θ is the angle between the chord and the x-axis.In our case, a = 1, so length is 4 / sin²θ.Given length is 4, so 4 = 4 / sin²θ => sin²θ = 1 => sinθ = ±1 => θ = 90° or 270°, which would make the line vertical, but our line is y = kx - k, which is not vertical unless k is infinite.This suggests that perhaps the length of the focal chord cannot be 4 unless the chord is vertical, but our line is not vertical.This is confusing.Alternatively, maybe I should use the parametric approach again.Given that t₁ and t₂ are roots of k t² - 2t - k = 0, so t₁ + t₂ = 2/k and t₁ t₂ = -1.Wait, from quadratic equation, t₁ + t₂ = 2/k and t₁ t₂ = -k/k = -1.So, t₁ t₂ = -1.Now, the distance between points A and B can be found using the parametric distance formula.In parametric form, the distance between two points t₁ and t₂ on the parabola y² = 4ax is given by:Distance = sqrt[(x₁ - x₂)^2 + (y₁ - y₂)^2]Where x₁ = t₁², y₁ = 2t₁, x₂ = t₂², y₂ = 2t₂.So,Distance = sqrt[(t₁² - t₂²)^2 + (2t₁ - 2t₂)^2]Factor:= sqrt[(t₁ - t₂)^2(t₁ + t₂)^2 + 4(t₁ - t₂)^2]Factor out (t₁ - t₂)^2:= sqrt[(t₁ - t₂)^2 [ (t₁ + t₂)^2 + 4 ] ]= |t₁ - t₂| sqrt[ (t₁ + t₂)^2 + 4 ]Given that the distance is 4, so:|t₁ - t₂| sqrt[ (t₁ + t₂)^2 + 4 ] = 4We know from quadratic equation that t₁ + t₂ = 2/k and t₁ t₂ = -1.We can find |t₁ - t₂| using the identity:(t₁ - t₂)^2 = (t₁ + t₂)^2 - 4 t₁ t₂= (2/k)^2 - 4*(-1)= 4/k² + 4= 4(1/k² + 1)So, |t₁ - t₂| = 2 sqrt(1/k² + 1) = 2 sqrt( (1 + k²)/k² ) = 2 sqrt(1 + k²)/kSo, plug back into distance equation:[2 sqrt(1 + k²)/k] * sqrt[ (2/k)^2 + 4 ] = 4Simplify inside the second sqrt:(2/k)^2 + 4 = 4/k² + 4 = 4(1/k² + 1) = 4(1 + k²)/k²So, sqrt[4(1 + k²)/k²] = 2 sqrt(1 + k²)/kTherefore, distance equation becomes:[2 sqrt(1 + k²)/k] * [2 sqrt(1 + k²)/k] = 4Multiply:4(1 + k²)/k² = 4Divide both sides by 4:(1 + k²)/k² = 1Multiply both sides by k²:1 + k² = k²Subtract k²:1 = 0Again, this is impossible. Hmm, I must be missing something.Wait, maybe the parametric approach is leading me to the same contradiction, which suggests that perhaps the line cannot intersect the parabola at two points with distance 4? But the problem states that it does.Alternatively, perhaps I made a mistake in the parametric distance formula.Wait, let me double-check the distance formula for two points on the parabola.Given points A(t₁², 2t₁) and B(t₂², 2t₂), the distance AB is:sqrt[(t₁² - t₂²)^2 + (2t₁ - 2t₂)^2]= sqrt[(t₁ - t₂)^2(t₁ + t₂)^2 + 4(t₁ - t₂)^2]= |t₁ - t₂| sqrt[(t₁ + t₂)^2 + 4]Yes, that seems correct.Given that, and knowing t₁ + t₂ = 2/k and t₁ t₂ = -1, we derived |t₁ - t₂| = 2 sqrt(1 + k²)/k.Then, the distance becomes:[2 sqrt(1 + k²)/k] * sqrt[(2/k)^2 + 4] = 4Which simplifies to:[2 sqrt(1 + k²)/k] * [2 sqrt(1 + k²)/k] = 4Which is 4(1 + k²)/k² = 4Thus, (1 + k²)/k² = 1 => 1 + k² = k² => 1 = 0, which is impossible.This suggests that there is no such k where the distance between A and B is 4. But the problem states that such points exist, so I must have made a wrong assumption.Wait, perhaps the line is not a focal chord? But the line passes through (1,0), which is the focus, so it should be a focal chord.Alternatively, maybe I made a mistake in the parametric equations.Wait, in the parametric form, for parabola y² = 4ax, the parametric equations are x = at², y = 2at. In our case, a = 1, so x = t², y = 2t. That's correct.So, substituting into the line equation y = kx - k:2t = k t² - kWhich gives k t² - 2t - k = 0. Correct.Then, t₁ + t₂ = 2/k, t₁ t₂ = -1. Correct.Then, |t₁ - t₂| = sqrt[(t₁ + t₂)^2 - 4 t₁ t₂] = sqrt[(4/k²) - 4*(-1)] = sqrt[4/k² + 4] = 2 sqrt(1/k² + 1) = 2 sqrt( (1 + k²)/k² ) = 2 sqrt(1 + k²)/k. Correct.Then, distance AB = |t₁ - t₂| sqrt[(t₁ + t₂)^2 + 4] = [2 sqrt(1 + k²)/k] * sqrt[(4/k²) + 4] = [2 sqrt(1 + k²)/k] * [2 sqrt(1 + k²)/k] = 4(1 + k²)/k². Correct.Set equal to 4:4(1 + k²)/k² = 4 => (1 + k²)/k² = 1 => 1 + k² = k² => 1 = 0. Contradiction.This suggests that there is no real k for which the distance AB is 4. But the problem says it does, so perhaps I made a wrong assumption.Wait, maybe the line is not a focal chord? But the line passes through (1,0), which is the focus, so it must be a focal chord.Alternatively, perhaps the distance AB is not 4, but the problem says it is. So, maybe I need to reconsider.Wait, perhaps the line is not intersecting the parabola at two distinct points, but the problem says it does. So, maybe the distance is 4, but my calculation is wrong.Alternatively, maybe I should approach this differently.Let me consider the midpoint of AB.Let me denote the midpoint as M.Coordinates of M: ( (x₁ + x₂)/2, (y₁ + y₂)/2 )From the quadratic equation, the sum of roots x₁ + x₂ = (2k² + 4)/k²So, x-coordinate of M: (x₁ + x₂)/2 = (2k² + 4)/(2k²) = (k² + 2)/k²Similarly, y-coordinate of M: (y₁ + y₂)/2From earlier, y₁ + y₂ = [2(1 + sqrt(k² + 1))]/k + [2(1 - sqrt(k² + 1))]/k = [2 + 2 sqrt(k² + 1) + 2 - 2 sqrt(k² + 1)] /k = 4/kSo, y-coordinate of M: (4/k)/2 = 2/kSo, midpoint M is at ( (k² + 2)/k² , 2/k )Now, the distance from M to the y-axis is the absolute value of its x-coordinate, which is |(k² + 2)/k²| = (k² + 2)/k², since k² is positive.So, distance is (k² + 2)/k² = 1 + 2/k²Now, the problem asks for this distance, given that AB = 4.But from earlier, we saw that AB = 4 leads to a contradiction, suggesting that perhaps the distance is fixed regardless of k.Wait, but that can't be, because the distance from M to y-axis is 1 + 2/k², which depends on k.But since AB = 4 leads to a contradiction, perhaps the only way this can happen is if the line is such that the chord AB is of length 4, but that's impossible, so maybe the distance from M to y-axis is fixed regardless.Wait, but the problem gives options, so perhaps the distance is fixed.Wait, let me think differently.Since the line passes through the focus (1,0), and the parabola is y² = 4x, the midpoint M of AB lies on the axis of the parabola, which is the x-axis. Wait, no, the axis of the parabola is the x-axis, but the midpoint M is not necessarily on the axis.Wait, but in our earlier calculation, the y-coordinate of M is 2/k, which is not necessarily zero unless k approaches infinity.Wait, perhaps the midpoint lies on the directrix? The directrix is x = -1.Wait, the distance from M to the y-axis is x-coordinate of M, which is (k² + 2)/k².But I need to find this distance given that AB = 4.But since AB = 4 leads to a contradiction, perhaps the only way is that the distance is fixed.Wait, let me think about the properties of the parabola.In a parabola, the midpoint of a focal chord lies on the axis of the parabola. Wait, is that true?Wait, no. The midpoint of a focal chord does not necessarily lie on the axis. For example, if the chord is vertical, the midpoint would lie on the axis, but for other chords, it doesn't.Wait, but in our case, the line passes through the focus, so it's a focal chord, but the midpoint is not necessarily on the axis.Wait, but let me check.From our earlier calculation, the midpoint M is at ( (k² + 2)/k² , 2/k )So, unless 2/k = 0, which would require k to be infinite, the midpoint is not on the x-axis.Wait, but maybe there's another property.Wait, in a parabola, the midpoint of a focal chord has a special relation with the directrix.Wait, the directrix is x = -1.The distance from M to the directrix is |x_M - (-1)| = |(k² + 2)/k² + 1| = |(k² + 2 + k²)/k²| = |2k² + 2| /k² = 2(k² + 1)/k²But from earlier, the distance AB is 4(k² + 1)/k², which is given as 4.So, 4(k² + 1)/k² = 4 => (k² + 1)/k² = 1 => 1 + 1/k² = 1 => 1/k² = 0 => k² approaches infinity, which would mean k approaches infinity, making the line vertical.But a vertical line through (1,0) is x =1, which intersects the parabola y² = 4x at y² =4*1=4, so y=±2, so points (1,2) and (1,-2). The distance AB is sqrt[(1-1)^2 + (2 - (-2))^2] = sqrt[0 + 16] =4. So, yes, in this case, the distance AB is 4.So, when k approaches infinity, the line becomes x=1, which is a vertical line through the focus, intersecting the parabola at (1,2) and (1,-2), with AB=4.In this case, the midpoint M is at (1,0), which is the focus.So, the distance from M to the y-axis is |1| =1.Therefore, the answer is 1.But wait, in this case, the line is x=1, which is a vertical line, and k approaches infinity, so in our earlier parametrization, k is infinite, so t₁ and t₂ would be [1 ± sqrt(1 + k²)] /k ≈ [1 ± k]/k, so t₁ ≈ (1 +k)/k ≈1 +1/k ≈1, and t₂≈(1 -k)/k≈-1.So, points A and B would be (1,2) and (1,-2), which matches.So, in this case, the midpoint is (1,0), distance to y-axis is 1.Therefore, the answer is 1, which is option B.But wait, earlier, when I tried to compute using finite k, I got a contradiction, but when k approaches infinity, it works.So, the distance from the midpoint to the y-axis is 1.Therefore, the answer is B:1.

answer：First, I need to understand the problem. We have a list of nine positive integers, and six of them are given: 4, 5, 3, 7, 9, and 6. The goal is to determine the largest possible value of the median of this list.The median of a list of nine numbers is the fifth number when the list is sorted in ascending order. To maximize the median, I should aim to make the fifth number as large as possible.I'll start by sorting the given six numbers: 3, 4, 5, 6, 7, 9.Since we need to add three more numbers to make a total of nine, I can choose these numbers strategically. To maximize the median, I should add numbers that are larger than the current largest number in the list, which is 9. For example, I can add 10, 11, and 12.Adding these numbers, the complete sorted list becomes: 3, 4, 5, 6, 7, 9, 10, 11, 12.Now, the fifth number in this sorted list is 7, which is the median.Therefore, the largest possible value of the median is 7.

answer：First, I recognize that the sets ({a, 1}) and ({0, a + b}) are equal. This means that every element in one set must also be in the other set and vice versa.Since (0) is an element of the second set, it must also be an element of the first set. Therefore, (a) must be equal to (0).Next, since the sets are equal, the other element (1) in the first set must correspond to (a + b) in the second set. Substituting (a = 0) into this equation gives:[0 + b = 1 implies b = 1]Finally, to find (b - a), I substitute the values of (a) and (b):[b - a = 1 - 0 = 1]