answer：First, I'll break down the expression ( lg25 + 2lg2 + 8^{frac{2}{3}} ) into its individual components to simplify each part.For the logarithmic terms, I recognize that ( lg25 ) can be rewritten using the property ( lg a^b = b lg a ). Since ( 25 = 5^2 ), this becomes ( 2lg5 ).Next, the term ( 2lg2 ) is already in a simplified form, so I'll keep it as is.For the exponential term ( 8^{frac{2}{3}} ), I can express 8 as ( 2^3 ). Applying the exponent rule ( (a^m)^n = a^{m cdot n} ), this becomes ( (2^3)^{frac{2}{3}} = 2^{2} = 4 ).Now, I'll combine the simplified logarithmic terms. Since ( lg5 + lg2 = lg(5 times 2) = lg10 ), and knowing that ( lg10 = 1 ), the expression simplifies to ( 2 times 1 = 2 ).Finally, I'll add the result of the logarithmic terms to the exponential term: ( 2 + 4 = 6 ).

answer：Okay, so I have this problem where I need to find the number of elements in the set P + Q. P is given as {0, 2, 5} and Q is {1, 2, 6}. Hmm, let me make sure I understand what P + Q means. It says that P + Q is the set of all sums a + b where a is in P and b is in Q. So, basically, I need to add every element of P to every element of Q and then count how many unique results I get.Alright, let's break this down. I think the best way is to list out all possible sums. Since P has 3 elements and Q has 3 elements, there should be 3 times 3, which is 9 possible sums. But some of these sums might be the same, so the actual number of unique elements in P + Q might be less than 9.Let me write down all the possible combinations:Starting with the first element of P, which is 0:- 0 + 1 = 1- 0 + 2 = 2- 0 + 6 = 6Next, the second element of P, which is 2:- 2 + 1 = 3- 2 + 2 = 4- 2 + 6 = 8Then, the third element of P, which is 5:- 5 + 1 = 6- 5 + 2 = 7- 5 + 6 = 11Okay, so now let me list all these results: 1, 2, 6, 3, 4, 8, 6, 7, 11.Wait, I see that 6 appears twice. So, when I collect all the unique elements, I need to remove duplicates. Let me write them out in order to check for duplicates:1, 2, 3, 4, 6, 7, 8, 11.Hmm, let me count these. 1, 2, 3, 4, 6, 7, 8, 11. That's 8 numbers. So, does that mean there are 8 unique elements in P + Q?Wait, let me double-check to make sure I didn't miss any duplicates. So, going back through the sums:From 0 + Q: 1, 2, 6.From 2 + Q: 3, 4, 8.From 5 + Q: 6, 7, 11.So, the sums are 1, 2, 6, 3, 4, 8, 6, 7, 11. So, 6 is repeated once, so that's the only duplicate. So, when I remove duplicates, I have 8 unique numbers.Wait, let me count again: 1, 2, 3, 4, 6, 7, 8, 11. Yes, that's 8 numbers. So, the number of elements in P + Q is 8.But just to make sure, let me think if there's another way to approach this. Maybe by considering the possible ranges or something. Let's see.The smallest element in P is 0, and the smallest in Q is 1, so the smallest sum is 0 + 1 = 1.The largest element in P is 5, and the largest in Q is 6, so the largest sum is 5 + 6 = 11.So, the possible sums range from 1 to 11. Now, let's see which numbers in between are missing.Looking at the sums we have: 1, 2, 3, 4, 6, 7, 8, 11.So, missing numbers between 1 and 11 are 5, 9, 10.Wait, is 5 missing? Let me check if 5 can be formed by any sum. Let's see:Is there a pair a in P and b in Q such that a + b = 5?Looking at P: 0, 2, 5.Looking at Q: 1, 2, 6.0 + 5? But 5 isn't in Q. 2 + 3? 3 isn't in Q. 5 + 0? 0 isn't in Q. So, no, 5 can't be formed.Similarly, 9: Let's see if 9 is possible.Looking for a + b = 9.Possible a's: 0, 2, 5.0 + 9? 9 isn't in Q.2 + 7? 7 isn't in Q.5 + 4? 4 isn't in Q.So, 9 can't be formed.10: Let's check if 10 can be formed.a + b = 10.0 + 10? 10 isn't in Q.2 + 8? 8 isn't in Q.5 + 5? 5 isn't in Q.So, 10 can't be formed either.Therefore, the sums we have are 1, 2, 3, 4, 6, 7, 8, 11, which are 8 numbers, and 5, 9, 10 are missing. So, that confirms that P + Q has 8 elements.Just to recap, I listed all possible sums, removed duplicates, and then checked the range to see if any numbers were missing. It seems that 5, 9, and 10 are not achievable, so we have 8 unique sums.I think that's thorough enough. I don't see any mistakes in my reasoning. So, the number of elements in P + Q is 8.**Final Answer**The number of elements in ( P + Q ) is boxed{8}.

answer：Alright, so I have this problem here: "How many three-digit numbers are increased by 99 when their digits are reversed?" The options are A) 4, B) 8, C) 10, D) 80, E) 90. Hmm, okay, let me try to figure this out step by step.First, let's understand the problem. We need to find all three-digit numbers such that when you reverse their digits, the new number is 99 more than the original number. So, if the original number is ABC (where A, B, C are digits), then reversing it gives CBA, and we have CBA = ABC + 99.Let me write that down algebraically. Let the original number be 100*A + 10*B + C. When reversed, it becomes 100*C + 10*B + A. According to the problem, reversing the digits increases the number by 99, so:100*C + 10*B + A = 100*A + 10*B + C + 99Okay, so let me simplify this equation. Subtract 100*A + 10*B + C from both sides:100*C + 10*B + A - (100*A + 10*B + C) = 99Simplify the left side:100C + 10B + A - 100A - 10B - C = 99Combine like terms:(100C - C) + (10B - 10B) + (A - 100A) = 99So that's:99C - 99A = 99Hmm, factor out 99:99*(C - A) = 99Divide both sides by 99:C - A = 1So, C = A + 1Alright, so the units digit is one more than the hundreds digit. That's a key relationship.Now, let's recall that A, B, C are digits. Since it's a three-digit number, A can't be zero. So A is from 1 to 9, and C is from 0 to 9. But since C = A + 1, C must be at least 1 (since A is at least 1). So A can be from 1 to 8 because if A is 9, then C would be 10, which isn't a digit. So A is 1 through 8, and C is 2 through 9.What about B? In the equation above, B cancels out. So B can be any digit from 0 to 9. That means for each valid A (from 1 to 8), B can be 0-9, giving 10 possibilities each.So the number of such three-digit numbers is 8 (choices for A) * 10 (choices for B) = 80.Wait, but hold on. Let me double-check that. Because sometimes when you reverse digits, there might be some constraints I haven't considered. For example, does reversing the digits always result in a valid three-digit number? Well, since the original number is three-digit, reversing it will also be three-digit because C can't be zero (since C = A + 1, and A is at least 1, so C is at least 2). So, the reversed number is also a three-digit number, so that's fine.Another thing to check: when you reverse the digits, does the difference always come out to exactly 99? Let me test with an example.Take A=1, B=0, C=2. Original number is 102, reversed is 201. 201 - 102 = 99. That works.Another example: A=2, B=5, C=3. Original number is 253, reversed is 352. 352 - 253 = 99. Yep, that works too.What about A=8, B=9, C=9. Original number is 899, reversed is 998. 998 - 899 = 99. Perfect.So, it seems that for each A from 1 to 8, and any B from 0 to 9, the condition holds. So that's 8*10=80 numbers.Looking back at the options, D is 80, so that should be the answer.Wait, but hold on a second. Let me make sure I didn't miss any constraints. The problem says "three-digit numbers," so A can't be zero, which we've already considered. Also, since C = A + 1, C is always a digit because A is at most 8, so C is at most 9. So, no issues there.Is there any case where reversing the digits doesn't increase the number by exactly 99? Let me see. Suppose A=1, B=9, C=2. Original number is 192, reversed is 291. 291 - 192 = 99. Still works.Another edge case: A=1, B=0, C=2: 102 reversed is 201, difference is 99. A=8, B=0, C=9: 809 reversed is 908, 908 - 809 = 99. So, yes, all these cases work.Therefore, I think my initial conclusion is correct. There are 80 such numbers.**Final Answer**boxed{D}

answer：First, determine the total number of phrases the parrot knows now, which is 17.Next, subtract the 3 phrases the parrot already knew when Georgina bought it. This leaves 14 new phrases learned.Since Georgina teaches the parrot 2 phrases each week, divide the number of new phrases by 2 to find the number of weeks: 14 ÷ 2 = 7 weeks.Finally, convert the number of weeks to days by multiplying by 7: 7 × 7 = 49 days.Therefore, Georgina has had the parrot for 49 days.

answer：Okay, so I have this problem about triangle ABC with sides AB = 25, BC = 39, and AC = 42. Points D and E are on AB and AC respectively, with AD = 19 and AE = 14. I need to find the ratio of the area of triangle ADE to the area of quadrilateral BCED. The answer choices are given, so that might help if I get stuck or want to check my answer.First, I should probably visualize the triangle ABC. Let me sketch it out in my mind. AB is 25 units, BC is 39, and AC is 42. So, it's a scalene triangle since all sides are different. Points D and E are on AB and AC. AD is 19, so since AB is 25, that means DB is 25 - 19 = 6. Similarly, AE is 14, so EC is 42 - 14 = 28.I think the key here is to find the areas of triangle ADE and quadrilateral BCED. Then, take the ratio of those two areas. Since the problem is about ratios, maybe I don't need to calculate the actual areas but can find the ratio using some proportional reasoning.I remember that in triangles, if two sides are divided proportionally, the areas of the resulting smaller triangles are proportional to the product of the segments. So, maybe I can use the concept of similar triangles or area ratios based on segment ratios.Let me recall the formula for the area of a triangle. Area = (1/2)*base*height. But since I don't know the heights here, maybe I can use another approach. Perhaps using Heron's formula to find the area of triangle ABC first, and then find the areas of ADE and BCED based on that.Wait, Heron's formula requires knowing all three sides, which I do for triangle ABC. Let me compute the semi-perimeter (s) first.s = (AB + BC + AC)/2 = (25 + 39 + 42)/2 = (106)/2 = 53.Then, the area of triangle ABC is sqrt[s(s - AB)(s - BC)(s - AC)] = sqrt[53*(53 - 25)*(53 - 39)*(53 - 42)].Calculating each term:53 - 25 = 28,53 - 39 = 14,53 - 42 = 11.So, area = sqrt[53*28*14*11].Let me compute that step by step.First, multiply 53 and 28: 53*28. Hmm, 50*28 = 1400, 3*28=84, so total is 1400 + 84 = 1484.Then, multiply 14 and 11: 14*11 = 154.Now, multiply 1484 and 154. That's a bit more involved. Let me break it down:1484 * 154 = 1484*(150 + 4) = 1484*150 + 1484*4.1484*150: 1484*100 = 148,400; 1484*50 = 74,200. So, total is 148,400 + 74,200 = 222,600.1484*4 = 5,936.So, total is 222,600 + 5,936 = 228,536.Therefore, the area is sqrt[228,536]. Let me see if I can compute that.Wait, 228,536 is a large number. Maybe I made a mistake in calculation because Heron's formula can sometimes result in large numbers, but let me check.Wait, no, Heron's formula is correct, but perhaps I can factor it differently.Wait, 53*28*14*11. Let me factor each number:53 is prime.28 = 4*7,14 = 2*7,11 is prime.So, putting it all together: 53 * (4*7) * (2*7) * 11.So, that's 53 * 4 * 7 * 2 * 7 * 11.Grouping the numbers:4*2 = 8,7*7 = 49,So, 8 * 49 * 53 * 11.Compute 8*49 = 392,Then, 392 * 53. Let me compute 392*50 = 19,600 and 392*3 = 1,176, so total is 19,600 + 1,176 = 20,776.Then, 20,776 * 11. That's 20,776*10 = 207,760 + 20,776 = 228,536. So, same as before.So, sqrt(228,536). Let me see if this is a perfect square.Let me try to find the square root. Let's see, 478^2 = 478*478.Compute 400^2 = 160,000,70^2 = 4,900,8^2 = 64,But 478^2 = (400 + 78)^2 = 400^2 + 2*400*78 + 78^2 = 160,000 + 62,400 + 6,084 = 160,000 + 62,400 = 222,400 + 6,084 = 228,484.Hmm, 478^2 = 228,484, which is less than 228,536 by 52. So, 478^2 = 228,484, 479^2 = 478^2 + 2*478 +1 = 228,484 + 956 + 1 = 229,441. That's higher than 228,536.So, sqrt(228,536) is between 478 and 479. So, approximately 478.5 or something. But since we need an exact value, maybe I made a mistake in the approach.Wait, perhaps Heron's formula isn't the best approach here because the numbers are messy. Maybe instead of calculating the area of ABC, I can find the ratio of areas using the concept of similar triangles or area ratios based on the segments.I remember that if two triangles share the same angle, the ratio of their areas is equal to the product of the ratios of their sides. So, in this case, triangle ADE and triangle ABC share angle A. So, the ratio of their areas should be (AD/AB)*(AE/AC).Let me write that down:Area ratio (ADE/ABC) = (AD/AB)*(AE/AC).Plugging in the values:AD = 19, AB = 25,AE = 14, AC = 42.So, Area ratio = (19/25)*(14/42).Simplify 14/42: that's 1/3.So, Area ratio = (19/25)*(1/3) = 19/75.So, the area of triangle ADE is 19/75 times the area of triangle ABC.Therefore, the area of quadrilateral BCED would be the area of ABC minus the area of ADE, which is (1 - 19/75) times the area of ABC.Compute 1 - 19/75 = (75 - 19)/75 = 56/75.So, the area of BCED is 56/75 times the area of ABC.Therefore, the ratio of the area of ADE to BCED is (19/75)/(56/75) = 19/56.Wait, that's one of the answer choices. Option D is 19/56. So, is that the answer?Wait, hold on. Let me double-check my reasoning.I used the formula that the ratio of areas of two triangles sharing a common angle is equal to the product of the ratios of their sides. So, that is, for triangles ADE and ABC, which share angle A, the ratio of their areas is (AD/AB)*(AE/AC). That seems correct.So, plugging in the numbers: (19/25)*(14/42) = (19/25)*(1/3) = 19/75. So, area of ADE is 19/75 of ABC.Therefore, the area of quadrilateral BCED is ABC - ADE = 1 - 19/75 = 56/75 of ABC.Therefore, ratio ADE : BCED is (19/75) : (56/75) = 19:56, so 19/56.So, the answer should be D) 19/56.Wait, but let me think again. Is this correct? Because sometimes when dealing with areas in triangles, especially when points are on sides, the ratio can be tricky.Alternatively, maybe I can use coordinate geometry to compute the areas.Let me assign coordinates to the triangle ABC. Let me place point A at (0,0), point B at (25,0), and point C somewhere in the plane. Then, I can compute the coordinates of D and E, and then compute the areas.But that might be more involved, but let's try.First, set A at (0,0). Let me denote point B as (25,0). Then, point C is somewhere in the plane. We need to find its coordinates.Given that AC = 42 and BC = 39. So, coordinates of C satisfy the distances from A and B.So, coordinates of C = (x,y). Then, distance from A is sqrt(x^2 + y^2) = 42, so x^2 + y^2 = 42^2 = 1764.Distance from B is sqrt((x - 25)^2 + y^2) = 39, so (x - 25)^2 + y^2 = 39^2 = 1521.Subtracting the two equations:(x - 25)^2 + y^2 - (x^2 + y^2) = 1521 - 1764.Expanding (x - 25)^2: x^2 - 50x + 625.So, x^2 - 50x + 625 + y^2 - x^2 - y^2 = -243.Simplify: -50x + 625 = -243.So, -50x = -243 - 625 = -868.Thus, x = (-868)/(-50) = 868/50 = 17.36.So, x = 17.36. Then, plug back into x^2 + y^2 = 1764.Compute x^2: 17.36^2. Let me compute 17^2 = 289, 0.36^2 = 0.1296, and cross term 2*17*0.36 = 12.24.So, (17 + 0.36)^2 = 17^2 + 2*17*0.36 + 0.36^2 = 289 + 12.24 + 0.1296 ≈ 301.3696.So, x^2 ≈ 301.3696.Therefore, y^2 = 1764 - 301.3696 ≈ 1462.6304.So, y ≈ sqrt(1462.6304). Let me compute sqrt(1462.6304).Well, 38^2 = 1444, 39^2 = 1521. So, sqrt(1462.6304) is between 38 and 39.Compute 38.25^2: 38^2 + 2*38*0.25 + 0.25^2 = 1444 + 19 + 0.0625 = 1463.0625.That's very close to 1462.6304. So, y ≈ 38.25 - a little bit.Compute 38.25^2 = 1463.0625.Difference: 1463.0625 - 1462.6304 = 0.4321.So, approximate y ≈ 38.25 - (0.4321)/(2*38.25) ≈ 38.25 - 0.4321/76.5 ≈ 38.25 - 0.0056 ≈ 38.2444.So, y ≈ 38.2444.So, coordinates of C are approximately (17.36, 38.2444).So, now, points D and E.Point D is on AB, with AD = 19. Since AB is from (0,0) to (25,0), so D is at (19, 0).Point E is on AC, with AE = 14. Since AC is from (0,0) to (17.36, 38.2444). So, we can parametrize AC.Parametric equations:x = (17.36/42)*t,y = (38.2444/42)*t,where t ranges from 0 to 42.Since AE = 14, t = 14.So, coordinates of E:x = (17.36/42)*14 = (17.36)*(14/42) = 17.36*(1/3) ≈ 5.7867,y = (38.2444/42)*14 = (38.2444)*(14/42) = 38.2444*(1/3) ≈ 12.7481.So, E is approximately at (5.7867, 12.7481).Now, we have coordinates for A(0,0), D(19,0), E(5.7867,12.7481).We can compute the area of triangle ADE using the shoelace formula.Shoelace formula:Area = (1/2)|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.Plugging in A(0,0), D(19,0), E(5.7867,12.7481):Area = (1/2)|0*(0 - 12.7481) + 19*(12.7481 - 0) + 5.7867*(0 - 0)|Simplify:= (1/2)|0 + 19*12.7481 + 0|= (1/2)*(19*12.7481)Compute 19*12.7481:10*12.7481 = 127.481,9*12.7481 = 114.7329,Total: 127.481 + 114.7329 = 242.2139.So, area ≈ (1/2)*242.2139 ≈ 121.10695.So, area of ADE is approximately 121.11.Now, let's compute the area of quadrilateral BCED. Since BCED is a quadrilateral, perhaps I can compute it as the area of ABC minus the area of ADE.Earlier, using Heron's formula, I got the area of ABC as sqrt(228,536) ≈ 478.05.Wait, but earlier, when I tried Heron's formula, I had an approximate value of 478.05, but when I computed using coordinates, the area of ABC can be found as well.Wait, using coordinates, the area of ABC can be found via shoelace formula.Coordinates of A(0,0), B(25,0), C(17.36,38.2444).Shoelace formula:Area = (1/2)|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|Plugging in:= (1/2)|0*(0 - 38.2444) + 25*(38.2444 - 0) + 17.36*(0 - 0)|= (1/2)|0 + 25*38.2444 + 0|= (1/2)*(956.11)≈ (1/2)*956.11 ≈ 478.055.So, area of ABC is approximately 478.055.Earlier, area of ADE is approximately 121.11.So, area of BCED is 478.055 - 121.11 ≈ 356.945.Therefore, the ratio of ADE to BCED is 121.11 / 356.945 ≈ 0.339, which is approximately 1/3.Wait, but 121.11 / 356.945 is approximately 0.339, which is roughly 1/3. But 19/56 is approximately 0.339 as well because 19 divided by 56 is approximately 0.339.Wait, 19/56 is exactly 0.339285714..., which is approximately 0.339. So, that matches.So, both methods give me the same ratio, 19/56.Therefore, the answer is D) 19/56.Alternatively, another approach is to use vectors or mass point geometry, but I think the area ratio method is sufficient here.Wait, another thought: Since we have points D on AB and E on AC, the ratio of areas can also be found using the formula:Area ADE / Area ABC = (AD/AB) * (AE/AC).Which is exactly what I did earlier, giving 19/25 * 14/42 = 19/75. Then, since the quadrilateral BCED is the remaining part, its area ratio is 1 - 19/75 = 56/75. Thus, the ratio ADE : BCED is 19/75 : 56/75 = 19:56.So, that's consistent as well.Therefore, I feel confident that the answer is D) 19/56.**Final Answer**boxed{D}

answer：Okay, so I have this problem here with two parts. Let me try to figure it out step by step. First, part (1): I need to find the equation of line ( l ) that passes through the intersection point ( P ) of lines ( l_1 ) and ( l_2 ), and is perpendicular to line ( l_3 ). Alright, let's start by finding the intersection point ( P ) of ( l_1 ) and ( l_2 ). The equations are:( l_1: 2x - y = 1 )( l_2: x + 2y = 3 )To find ( P ), I need to solve these two equations simultaneously. Maybe I can use substitution or elimination. Let me try elimination.From ( l_1 ), I can express ( y ) in terms of ( x ):( 2x - y = 1 Rightarrow y = 2x - 1 )Now, substitute this expression for ( y ) into ( l_2 ):( x + 2(2x - 1) = 3 )Let me compute that:( x + 4x - 2 = 3 )Combine like terms:( 5x - 2 = 3 )Add 2 to both sides:( 5x = 5 Rightarrow x = 1 )Now, substitute ( x = 1 ) back into ( y = 2x - 1 ):( y = 2(1) - 1 = 2 - 1 = 1 )So, the intersection point ( P ) is ( (1, 1) ). Got that.Next, I need to find the equation of line ( l ) that passes through ( P(1,1) ) and is perpendicular to ( l_3 ). The equation of ( l_3 ) is given as ( x - y + 1 = 0 ). Let me rewrite that in slope-intercept form to find its slope.Starting with ( x - y + 1 = 0 ):( -y = -x - 1 Rightarrow y = x + 1 )So, the slope of ( l_3 ) is 1. Since ( l ) is perpendicular to ( l_3 ), the slope of ( l ) must be the negative reciprocal of 1. The negative reciprocal of 1 is -1. So, the slope of ( l ) is -1.Now, with the slope of ( l ) being -1 and passing through ( (1,1) ), I can write the equation using point-slope form:( y - y_1 = m(x - x_1) )Plugging in the values:( y - 1 = -1(x - 1) )Simplify:( y - 1 = -x + 1 )Bring all terms to one side:( y - 1 + x - 1 = 0 Rightarrow x + y - 2 = 0 )Alternatively, I can write it as ( x + y = 2 ). So, that's the equation of line ( l ). Let me double-check my steps to make sure I didn't make a mistake.1. Found intersection point ( P(1,1) ) correctly by solving ( l_1 ) and ( l_2 ).2. Converted ( l_3 ) to slope-intercept form and found its slope is 1.3. Since ( l ) is perpendicular, slope is -1.4. Used point-slope form with point ( (1,1) ) and slope -1, leading to equation ( x + y = 2 ).Looks good. So, part (1) is done.Moving on to part (2): Line ( l ) is tangent to the circle ( C: (x - a)^2 + y^2 = 8 ) where ( a > 0 ). I need to find ( a ).First, let's recall that the condition for a line to be tangent to a circle is that the distance from the center of the circle to the line equals the radius of the circle.Given the circle ( C: (x - a)^2 + y^2 = 8 ), its center is at ( (a, 0) ) and the radius is ( sqrt{8} = 2sqrt{2} ).The equation of line ( l ) is ( x + y = 2 ). Let me write it in standard form: ( x + y - 2 = 0 ).The formula for the distance from a point ( (x_0, y_0) ) to the line ( Ax + By + C = 0 ) is:( text{Distance} = frac{|Ax_0 + By_0 + C|}{sqrt{A^2 + B^2}} )So, in our case, ( A = 1 ), ( B = 1 ), ( C = -2 ), and the center is ( (a, 0) ). Plugging into the distance formula:( text{Distance} = frac{|1 cdot a + 1 cdot 0 - 2|}{sqrt{1^2 + 1^2}} = frac{|a - 2|}{sqrt{2}} )Since the line is tangent to the circle, this distance must equal the radius ( 2sqrt{2} ). So,( frac{|a - 2|}{sqrt{2}} = 2sqrt{2} )Let me solve for ( |a - 2| ):Multiply both sides by ( sqrt{2} ):( |a - 2| = 2sqrt{2} times sqrt{2} = 2 times 2 = 4 )So, ( |a - 2| = 4 ). This gives two possibilities:1. ( a - 2 = 4 Rightarrow a = 6 )2. ( a - 2 = -4 Rightarrow a = -2 )But the problem states that ( a > 0 ), so ( a = 6 ) is the solution.Wait, let me verify this. If ( a = 6 ), the center is at ( (6, 0) ). The distance from ( (6, 0) ) to the line ( x + y - 2 = 0 ) is:( |6 + 0 - 2| / sqrt{1 + 1} = |4| / sqrt{2} = 4 / sqrt{2} = 2sqrt{2} ), which is equal to the radius. So, that's correct.If ( a = -2 ), the center is at ( (-2, 0) ). The distance would be:( |-2 + 0 - 2| / sqrt{2} = |-4| / sqrt{2} = 4 / sqrt{2} = 2sqrt{2} ), which is also equal to the radius. But since ( a > 0 ), we discard ( a = -2 ).Therefore, ( a = 6 ).Let me recap part (2):1. Identified the center and radius of the circle.2. Wrote the equation of line ( l ) in standard form.3. Applied the distance formula from center to line.4. Set the distance equal to radius and solved for ( a ).5. Discarded the negative solution since ( a > 0 ).Everything seems to check out.**Final Answer**(1) The equation of line ( l ) is boxed{x + y = 2}.(2) The value of ( a ) is boxed{6}.