💡First, I need to understand the ratio of the portions that Amanda, Ben, and Carlos share, which is given as 1:2:7. This means that for every unit of money Amanda receives, Ben receives twice as much, and Carlos receives seven times as much.Next, I know that Amanda's portion is 20. Since Amanda's share corresponds to 1 unit in the ratio, I can determine the value of each unit by dividing Amanda's portion by her ratio value: 20 ÷ 1 = 20 per unit.Now, I can calculate Ben's portion by multiplying his ratio value by the value of each unit: 2 units × 20 = 40.Similarly, Carlos's portion is calculated by multiplying his ratio value by the value of each unit: 7 units × 20 = 140.Finally, to find the total amount of money shared, I add up the portions of Amanda, Ben, and Carlos: 20 + 40 + 140 = 200.

💡First, I recognize that a standard deck of 52 cards is composed of 4 suits: two black suits (spades and clubs) and two red suits (hearts and diamonds). Each suit contains 13 cards.To determine the probability that the top card is either a heart or a diamond, I need to count the total number of red cards. Since there are 13 hearts and 13 diamonds, the total number of red cards is 26.Next, I calculate the probability by dividing the number of favorable outcomes (red cards) by the total number of possible outcomes (all cards in the deck). This gives me 26 divided by 52.Finally, I simplify the fraction 26/52 to its lowest terms, which is 1/2. Therefore, the probability that the top card is either a heart or a diamond is 1/2.

💡First, I need to determine the total sum of all 20 numbers. Since the average is 500, the total sum is 20 multiplied by 500, which equals 10,000.Next, I'll find the total sum of the 14 numbers with an average of 390. This is calculated by multiplying 14 by 390, resulting in 5,460.To find the sum of the remaining 6 numbers, I'll subtract the sum of the 14 numbers from the total sum of all 20 numbers: 10,000 minus 5,460 equals 4,540.Finally, to calculate the average of these 6 numbers, I'll divide the sum of 4,540 by 6, which gives me 756.67.

💡First, I need to determine the values of x, y, and z based on the information provided.1. **Calculating x:** - x is three-fourths the number of novels Bryan owns. - Bryan has 90 novels. - So, ( x = frac{3}{4} times 90 = 67.5 ).2. **Calculating y:** - y is 20% more than the number of comics in Bryan's collection. - Bryan has 160 comics. - So, ( y = 160 + (0.20 times 160) = 192 ).3. **Calculating z:** - z is half the number of rooms in Bryan's house. - Bryan has 12 rooms. - So, ( z = frac{12}{2} = 6 ).Next, I'll calculate the total number of books and magazines in each bookshelf.4. **Books per bookshelf:** - ( 27x = 27 times 67.5 = 1822.5 ).5. **Magazines per bookshelf:** - ( 80y = 80 times 192 = 15360 ).Now, I'll find the total number of books and magazines across all bookshelves.6. **Total books:** - ( 1822.5 times 23z = 1822.5 times 23 times 6 = 251,595 ).7. **Total magazines:** - ( 15360 times 23z = 15360 times 23 times 6 = 2,119,680 ).Finally, I'll add the total number of books and magazines to get the grand total.8. **Grand total:** - ( 251,595 + 2,119,680 = 2,371,275 ).Therefore, Bryan has a total of **2,371,275** books and magazines in his bookshelves.

💡Okay, so I have this problem about a moving circle that passes through a fixed point F(0,1) and is tangent to the line l₁: y = -1. The trajectory of the center of this circle is called E. I need to find the equation of this trajectory and then determine the maximum length of the chord PQ when a line l₂ intersects E at two points P and Q, with the midpoint of PQ having an ordinate (y-coordinate) of 2.Starting with part (1): Finding the equation of the trajectory of the center C.Hmm, so the circle passes through F(0,1) and is tangent to the line y = -1. Let me denote the center of the circle as C(h, k). Since the circle passes through F(0,1), the distance from C to F must be equal to the radius. Also, since the circle is tangent to the line y = -1, the distance from C to this line must also be equal to the radius.So, the radius r can be expressed in two ways:1. The distance from C(h, k) to F(0,1): r = √[(h - 0)² + (k - 1)²] = √(h² + (k - 1)²)2. The distance from C(h, k) to the line y = -1: Since the line is horizontal, the distance is just |k - (-1)| = |k + 1|Since both expressions equal the radius r, I can set them equal to each other:√(h² + (k - 1)²) = |k + 1|To eliminate the square root, I'll square both sides:h² + (k - 1)² = (k + 1)²Expanding both sides:h² + k² - 2k + 1 = k² + 2k + 1Subtracting k² and 1 from both sides:h² - 2k = 2kBringing all terms to one side:h² - 4k = 0So, h² = 4kThis is the equation of a parabola opening upwards with vertex at the origin. So, the trajectory E is the parabola x² = 4y.Wait, let me double-check that. If I set h² = 4k, then replacing h with x and k with y, the equation becomes x² = 4y, which is indeed a standard parabola. So, part (1) seems done.Moving on to part (2): Finding the maximum value of |PQ| where l₂ intersects E at P and Q, and the midpoint of PQ has a y-coordinate of 2.Alright, so the parabola is x² = 4y. Let me denote the line l₂ as having a slope m and passing through some point. But since we know the midpoint has a y-coordinate of 2, maybe it's better to approach this using the midpoint formula.Let me denote the midpoint as M(t, 2). So, the coordinates of M are (t, 2). Since M is the midpoint of PQ, the coordinates of P and Q can be expressed in terms of t and some variable related to the slope.Let me denote P as (x₁, y₁) and Q as (x₂, y₂). Then, since M is the midpoint:(x₁ + x₂)/2 = t and (y₁ + y₂)/2 = 2So, x₁ + x₂ = 2t and y₁ + y₂ = 4.Since both P and Q lie on the parabola x² = 4y, we have:x₁² = 4y₁ and x₂² = 4y₂Subtracting these two equations:x₁² - x₂² = 4(y₁ - y₂)Factor the left side:(x₁ - x₂)(x₁ + x₂) = 4(y₁ - y₂)We know that x₁ + x₂ = 2t, so:(x₁ - x₂)(2t) = 4(y₁ - y₂)Divide both sides by 2:(x₁ - x₂)t = 2(y₁ - y₂)But y₁ - y₂ can be expressed in terms of the slope of the line PQ. Let me denote the slope as m. Then, m = (y₁ - y₂)/(x₁ - x₂). So, y₁ - y₂ = m(x₁ - x₂)Substituting back:(x₁ - x₂)t = 2(m(x₁ - x₂))Assuming x₁ ≠ x₂ (since it's a chord, not a vertical line), we can divide both sides by (x₁ - x₂):t = 2mSo, m = t/2Therefore, the slope of the line PQ is t/2.Now, knowing the slope and the midpoint, we can write the equation of line PQ.Using point-slope form:y - 2 = (t/2)(x - t)So, the equation of l₂ is y = (t/2)x - (t²)/2 + 2Simplify:y = (t/2)x + (4 - t²)/2Now, to find the points P and Q where this line intersects the parabola x² = 4y, substitute y from the line equation into the parabola equation:x² = 4[(t/2)x + (4 - t²)/2]Simplify the right side:x² = 4*(t/2)x + 4*(4 - t²)/2x² = 2t x + 2*(4 - t²)x² = 2t x + 8 - 2t²Bring all terms to the left side:x² - 2t x - 8 + 2t² = 0So, the quadratic equation in x is:x² - 2t x + (2t² - 8) = 0Let me denote this as x² - 2t x + (2t² - 8) = 0Let me find the roots of this quadratic, which correspond to x₁ and x₂.Using the quadratic formula:x = [2t ± √( (2t)² - 4*1*(2t² - 8) )]/2Simplify inside the square root:(4t²) - 4*(2t² - 8) = 4t² - 8t² + 32 = -4t² + 32So,x = [2t ± √(-4t² + 32)]/2Factor out 4 from the square root:√(-4t² + 32) = √[4(-t² + 8)] = 2√(8 - t²)So,x = [2t ± 2√(8 - t²)]/2 = t ± √(8 - t²)Therefore, the x-coordinates of P and Q are x₁ = t + √(8 - t²) and x₂ = t - √(8 - t²)Now, let's find the corresponding y-coordinates using the parabola equation y = x²/4.So,y₁ = (t + √(8 - t²))² /4y₂ = (t - √(8 - t²))² /4Let me compute y₁ and y₂:First, expand (t + √(8 - t²))²:= t² + 2t√(8 - t²) + (8 - t²) = t² + 2t√(8 - t²) + 8 - t² = 8 + 2t√(8 - t²)Similarly, (t - √(8 - t²))²:= t² - 2t√(8 - t²) + (8 - t²) = t² - 2t√(8 - t²) + 8 - t² = 8 - 2t√(8 - t²)Therefore,y₁ = (8 + 2t√(8 - t²))/4 = 2 + (t√(8 - t²))/2y₂ = (8 - 2t√(8 - t²))/4 = 2 - (t√(8 - t²))/2Now, to find the distance |PQ|, we can use the distance formula:|PQ| = √[(x₁ - x₂)² + (y₁ - y₂)²]Compute x₁ - x₂:x₁ - x₂ = [t + √(8 - t²)] - [t - √(8 - t²)] = 2√(8 - t²)Similarly, y₁ - y₂:y₁ - y₂ = [2 + (t√(8 - t²))/2] - [2 - (t√(8 - t²))/2] = t√(8 - t²)Therefore,|PQ| = √[(2√(8 - t²))² + (t√(8 - t²))²]Compute each term:(2√(8 - t²))² = 4*(8 - t²) = 32 - 4t²(t√(8 - t²))² = t²*(8 - t²) = 8t² - t⁴So, adding them together:32 - 4t² + 8t² - t⁴ = 32 + 4t² - t⁴Therefore,|PQ| = √(32 + 4t² - t⁴)Hmm, that seems a bit complicated. Maybe I can factor this expression or find a way to maximize it.Let me denote the expression under the square root as f(t):f(t) = -t⁴ + 4t² + 32I need to find the maximum value of √(f(t)), which occurs when f(t) is maximized.So, let's find the maximum of f(t) = -t⁴ + 4t² + 32.This is a quartic function, but it's a downward opening quartic because the coefficient of t⁴ is negative. So, it will have a maximum.To find the maximum, take the derivative and set it to zero.f'(t) = -4t³ + 8tSet f'(t) = 0:-4t³ + 8t = 0Factor out -4t:-4t(t² - 2) = 0So, t = 0 or t² = 2 => t = ±√2Now, let's check the second derivative to ensure it's a maximum.f''(t) = -12t² + 8At t = 0:f''(0) = 8 > 0, which is a minimum.At t = ±√2:f''(√2) = -12*(2) + 8 = -24 + 8 = -16 < 0, which is a maximum.So, the maximum of f(t) occurs at t = ±√2.Compute f(√2):f(√2) = -( (√2)^4 ) + 4*(√2)^2 + 32= -(4) + 4*(2) + 32= -4 + 8 + 32= 36Similarly, f(-√2) is the same because all terms are even powers.So, the maximum value of f(t) is 36, which occurs at t = ±√2.Therefore, the maximum value of |PQ| is √36 = 6.Wait, let me just make sure I didn't make a mistake in computing f(t) at t = √2.Yes, (√2)^4 is (√2)^2 squared, which is 2 squared, so 4. Then, 4*(√2)^2 is 4*2=8. So, -4 + 8 +32 is indeed 36. So, √36 is 6.Therefore, the maximum length of |PQ| is 6.I think that's it. So, summarizing:1. The trajectory E is the parabola x² = 4y.2. The maximum length of |PQ| is 6.**Final Answer**The maximum value of (|PQ|) is (boxed{6}).

💡Okay, so I have this problem about a circle and a line, and I need to find the length of a tangent segment from a specific point to the circle. Let me try to break this down step by step.First, the circle is given by the equation (x - 4)² + (y + 2)² = 5. I remember that the standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. So, for this circle, the center must be at (4, -2), and the radius is the square root of 5, which is approximately 2.236. I'll just keep it as √5 for exactness.Next, the problem mentions a point M on the line y = x + 2. This point M is supposed to be the closest point on the line to the center of the circle. Hmm, okay, so I need to find the point on the line y = x + 2 that is closest to the center (4, -2). I think this is related to the concept of projecting a point onto a line. The closest point on a line to a given point is the perpendicular projection of that point onto the line.Let me recall the formula for the distance from a point to a line. The distance d from a point (x₀, y₀) to the line ax + by + c = 0 is given by:d = |a*x₀ + b*y₀ + c| / sqrt(a² + b²)But in this case, I don't just need the distance; I need the actual coordinates of the point M. So, maybe I should find the projection of the center onto the line y = x + 2.First, let me write the line in standard form. The given line is y = x + 2. To write it in ax + by + c = 0 form, I can subtract x and subtract 2 from both sides:-x + y - 2 = 0So, a = -1, b = 1, c = -2.Wait, actually, if I rearrange y = x + 2, it becomes x - y + 2 = 0. So, a = 1, b = -1, c = 2. That might be easier because the coefficients are positive. Let me confirm:Starting from y = x + 2, subtract x and subtract 2:y - x - 2 = 0, which is the same as -x + y - 2 = 0. So, a = -1, b = 1, c = -2.But I think it's more standard to have the x coefficient positive, so maybe I should write it as x - y + 2 = 0, which would make a = 1, b = -1, c = 2. Yeah, that seems better.So, the line is x - y + 2 = 0, with a = 1, b = -1, c = 2.Now, the formula for the distance from the center (4, -2) to this line is:d = |1*4 + (-1)*(-2) + 2| / sqrt(1² + (-1)²) = |4 + 2 + 2| / sqrt(1 + 1) = |8| / sqrt(2) = 8 / sqrt(2) = 4*sqrt(2)Wait, that seems a bit large. Let me double-check my calculations.Wait, the formula is |a*x₀ + b*y₀ + c| / sqrt(a² + b²). So plugging in:a = 1, x₀ = 4, b = -1, y₀ = -2, c = 2.So, 1*4 + (-1)*(-2) + 2 = 4 + 2 + 2 = 8. The absolute value is 8. The denominator is sqrt(1 + 1) = sqrt(2). So, 8 / sqrt(2) simplifies to 4*sqrt(2). Okay, that seems correct.But wait, the distance from the center to the line is 4*sqrt(2), which is approximately 5.656. But the radius of the circle is sqrt(5), which is about 2.236. So, the distance from the center to the line is much larger than the radius, which makes sense because the line is outside the circle.But the problem is asking for the length of the tangent segment from point M to the circle. I remember that the length of a tangent from a point to a circle is given by sqrt(d² - r²), where d is the distance from the point to the center, and r is the radius.Wait, but in this case, point M is on the line y = x + 2, and it's the closest point to the center. So, the distance from M to the center is the same as the distance from the center to the line, which is 4*sqrt(2). Therefore, the length of the tangent MN should be sqrt((4*sqrt(2))² - (sqrt(5))²).Let me compute that:(4*sqrt(2))² = 16*2 = 32(sqrt(5))² = 5So, sqrt(32 - 5) = sqrt(27) = 3*sqrt(3)So, the length of MN is 3*sqrt(3). That seems to be the answer.But wait, let me make sure I didn't skip any steps or make any mistakes. Let me go through it again.1. Circle C: (x - 4)² + (y + 2)² = 5. Center at (4, -2), radius sqrt(5).2. Line y = x + 2. Rewrite in standard form: x - y + 2 = 0.3. Find the point M on this line closest to the center (4, -2). This is the perpendicular projection.4. The distance from center to the line is |1*4 + (-1)*(-2) + 2| / sqrt(1 + 1) = |4 + 2 + 2| / sqrt(2) = 8 / sqrt(2) = 4*sqrt(2).5. The length of the tangent from M to the circle is sqrt(d² - r²) = sqrt((4*sqrt(2))² - (sqrt(5))²) = sqrt(32 - 5) = sqrt(27) = 3*sqrt(3).Yes, that seems consistent. I think I did everything correctly. The key steps were recognizing that the closest point M is the projection, calculating the distance from the center to the line, and then using the tangent length formula.I don't think I made any calculation errors. Let me just verify the distance formula one more time:For the line ax + by + c = 0 and point (x₀, y₀), distance is |a*x₀ + b*y₀ + c| / sqrt(a² + b²).Here, a = 1, b = -1, c = 2, x₀ = 4, y₀ = -2.So, 1*4 + (-1)*(-2) + 2 = 4 + 2 + 2 = 8. Absolute value is 8. Denominator is sqrt(1 + 1) = sqrt(2). So, 8 / sqrt(2) = 4*sqrt(2). Correct.Then, tangent length is sqrt(d² - r²) = sqrt(32 - 5) = sqrt(27) = 3*sqrt(3). Yep, that's right.I think I'm confident with this answer.