💡Alright, so I've got this problem about a hyperbola, and I need to find the value of 'a'. Let me try to recall what I know about hyperbolas and see if I can work through this step by step.First, the equation of the hyperbola is given as (x²/a²) - (y²/b²) = 1, where a and b are both positive. I remember that hyperbolas have asymptotes, which are straight lines that the hyperbola approaches but never touches. For this standard form of a hyperbola, the asymptotes are y = ±(b/a)x. So, the slopes of these asymptotes are ±(b/a).The problem states that one of the asymptotes has a slope of 2. Since the slopes are ±(b/a), that means either (b/a) = 2 or -(b/a) = 2. But since both a and b are positive, (b/a) must be positive, so (b/a) = 2. Therefore, b = 2a.Next, the problem mentions the focal length is 2√5. I think the focal length refers to the distance between the two foci of the hyperbola. For a hyperbola, the distance from the center to each focus is denoted by 'c', and the relationship between a, b, and c is given by c² = a² + b². So, the distance between the two foci is 2c, which is given as 2√5. That means c = √5.Now, since c² = a² + b², and we know that c = √5, then c² = (√5)² = 5. Also, from earlier, we have b = 2a, so b² = (2a)² = 4a².Substituting b² into the equation c² = a² + b², we get:5 = a² + 4a²5 = 5a²Dividing both sides by 5:1 = a²Taking the square root of both sides:a = √1a = 1Wait, that seems straightforward. Let me double-check to make sure I didn't make any mistakes.1. The asymptotes of the hyperbola (x²/a²) - (y²/b²) = 1 are y = ±(b/a)x. Given that one asymptote has a slope of 2, so (b/a) = 2, which gives b = 2a. That seems correct.2. The focal length is 2√5, so the distance from the center to each focus is √5. The relationship c² = a² + b² holds for hyperbolas, so plugging in c = √5, we get c² = 5.3. Substituting b = 2a into c² = a² + b² gives 5 = a² + 4a², which simplifies to 5 = 5a², leading to a² = 1, so a = 1. That makes sense.I don't see any errors in my reasoning. So, the value of 'a' must be 1.

💡Alright, so I've got this problem about two people, A and B, running around a circular track. They start from diametrically opposite positions, which means they're half a lap apart initially. They run at constant speeds but in opposite directions. The first time they meet, B has traveled 120 yards. The second time they meet, A is 80 yards short of completing one full lap. I need to find the circumference of the track.Okay, let's break this down. First, since they start diametrically opposite, the distance between them is half the circumference. Let's denote the circumference as C. So, the distance between A and B at the start is C/2.They start running at the same time and meet for the first time after B has traveled 120 yards. Since they're moving towards each other (because they're going in opposite directions), the sum of the distances they've covered when they meet for the first time should be equal to the initial distance between them, which is C/2.So, if B has traveled 120 yards, then A must have traveled (C/2 - 120) yards by the time they meet for the first time.Let's denote the speed of A as v_A and the speed of B as v_B. Since they started at the same time and met at the same time, the time taken for both to meet is the same. So, we can write:Time taken for first meeting = Distance traveled by A / Speed of A = (C/2 - 120) / v_ASimilarly, Time taken for first meeting = Distance traveled by B / Speed of B = 120 / v_BSince these times are equal, we have:(C/2 - 120) / v_A = 120 / v_BLet's denote this as Equation (1).Now, moving on to the second meeting. They meet again after some more time. By the time they meet the second time, A is 80 yards short of completing one full lap. That means A has traveled (C - 80) yards by the time they meet the second time.Similarly, let's figure out how much B has traveled by the second meeting. Since they meet again, the total distance covered by both A and B together should be equal to the circumference of the track. But wait, actually, since they're moving in opposite directions, each time they meet, the sum of their distances covered is equal to the circumference.Wait, actually, the first time they meet, they've covered a combined distance of C/2. The second time they meet, they would have covered another C distance, so total combined distance is C/2 + C = 3C/2.But let me think again. When two people are moving in opposite directions on a circular track, the time between consecutive meetings is the same, and the distance covered between meetings is equal to the circumference. So, the first meeting happens after they've covered C/2 together, and each subsequent meeting happens after they've covered another C together.Hmm, maybe I need to adjust my initial thought.Wait, no. Actually, when two people start moving towards each other on a circular track from diametrically opposite points, their first meeting occurs after they've covered half the circumference together. Then, each subsequent meeting occurs after they've covered the full circumference together. So, the first meeting: combined distance = C/2. The second meeting: combined distance = C/2 + C = 3C/2. The third meeting: combined distance = 3C/2 + C = 5C/2, and so on.So, in this problem, the second meeting occurs after they've covered a combined distance of 3C/2.Given that, let's figure out how much each has traveled by the second meeting.We know that by the second meeting, A has traveled (C - 80) yards. So, the distance traveled by B would be (3C/2 - (C - 80)) = (3C/2 - C + 80) = (C/2 + 80) yards.So, B has traveled (C/2 + 80) yards by the second meeting.Now, let's relate the distances traveled by A and B in the time between the first meeting and the second meeting.From the first meeting to the second meeting, the combined distance they cover is C. So, the distance A travels between the first and second meeting is (C - 80) - (C/2 - 120) = (C - 80 - C/2 + 120) = (C/2 + 40) yards.Similarly, the distance B travels between the first and second meeting is (C/2 + 80) - 120 = (C/2 - 40) yards.Since the time taken between the first and second meeting is the same for both A and B, the ratio of their speeds should be equal to the ratio of the distances they traveled in that time.So, v_A / v_B = (C/2 + 40) / (C/2 - 40)But from Equation (1), we have:(C/2 - 120) / v_A = 120 / v_BWhich can be rearranged to:v_A / v_B = (C/2 - 120) / 120So, we have two expressions for v_A / v_B:(C/2 + 40) / (C/2 - 40) = (C/2 - 120) / 120Now, let's solve this equation for C.Cross-multiplying:120 * (C/2 + 40) = (C/2 - 40) * (C/2 - 120)Let's compute both sides.Left side: 120*(C/2 + 40) = 60C + 4800Right side: (C/2 - 40)*(C/2 - 120) = (C/2)^2 - 120*(C/2) - 40*(C/2) + 40*120 = (C^2)/4 - 60C - 20C + 4800 = (C^2)/4 - 80C + 4800So, setting left side equal to right side:60C + 4800 = (C^2)/4 - 80C + 4800Subtract 4800 from both sides:60C = (C^2)/4 - 80CBring all terms to one side:(C^2)/4 - 80C - 60C = 0Simplify:(C^2)/4 - 140C = 0Multiply both sides by 4 to eliminate the fraction:C^2 - 560C = 0Factor out C:C(C - 560) = 0So, C = 0 or C = 560Since circumference can't be zero, C = 560 yards.Wait, but let me double-check my steps to make sure I didn't make a mistake.Starting from the equation:(C/2 + 40) / (C/2 - 40) = (C/2 - 120) / 120Cross-multiplying:120*(C/2 + 40) = (C/2 - 40)*(C/2 - 120)Left side: 60C + 4800Right side: (C^2)/4 - 80C + 4800So, 60C + 4800 = (C^2)/4 - 80C + 4800Subtract 4800: 60C = (C^2)/4 - 80CBring all terms to left: (C^2)/4 - 140C = 0Multiply by 4: C^2 - 560C = 0Factor: C(C - 560) = 0So, C = 560 yards.Yes, that seems correct. So, the circumference is 560 yards.

💡First, I need to simplify each part of the expression separately.For the first term, (3 times 3^4), I can use the product of powers property, which states that (a^m times a^n = a^{m+n}). Applying this property:[3 times 3^4 = 3^{1+4} = 3^5]Next, for the second term, (27^{63} div 27^{61}), I can use the quotient of powers property, which states that (a^m div a^n = a^{m-n}). Applying this property:[27^{63} div 27^{61} = 27^{63-61} = 27^2]Since (27) is equal to (3^3), I can rewrite (27^2) as:[27^2 = (3^3)^2 = 3^{6}]Now, the expression becomes:[3^5 - 3^6]Calculating the values of (3^5) and (3^6):[3^5 = 243][3^6 = 729]Subtracting these values:[243 - 729 = -486]Therefore, the final answer is (-486).

💡First, I need to determine the total number of marbles in the bag, which is 12 red marbles plus 8 blue marbles, totaling 20 marbles.Next, I'll calculate the total number of ways to draw 4 marbles out of 20 without replacement. This can be done using the combination formula ( binom{20}{4} ).Then, I'll find the number of ways to choose 2 red marbles out of the 12 available, which is ( binom{12}{2} ), and the number of ways to choose 2 blue marbles out of the 8 available, which is ( binom{8}{2} ).Multiplying the number of ways to choose the red and blue marbles gives the total number of favorable outcomes for selecting 2 red and 2 blue marbles.Finally, I'll divide the number of favorable outcomes by the total number of possible outcomes to find the probability of selecting exactly 2 red and 2 blue marbles.

💡Alright, so I've got this math problem here with four options labeled A, B, C, and D. Each option presents an equation, and I need to figure out which one is correct. Let's take them one by one.Starting with option A: ( a - 2a = a ). Hmm, okay, so if I have one 'a' and subtract two 'a's, what does that give me? Well, ( a - 2a ) is like saying I have one apple and then I take away two apples. That would leave me with negative one apple, right? So, ( a - 2a ) should equal ( -a ), not ( a ). So, option A doesn't seem correct.Moving on to option B: ( (-a^3 b)^2 = a^6 b^2 ). Alright, let's break this down. When you square a term, you square both the coefficient and the variables. Here, the term inside the parentheses is ( -a^3 b ). Squaring this means squaring the negative sign, the ( a^3 ), and the ( b ). Squaring the negative sign gives me a positive because ( (-1)^2 = 1 ). Squaring ( a^3 ) gives me ( a^{6} ) because ( (a^3)^2 = a^{3*2} = a^6 ). Similarly, squaring ( b ) gives me ( b^2 ). So, putting it all together, ( (-a^3 b)^2 ) equals ( a^6 b^2 ), which matches the right side of the equation. So, option B looks correct.But just to be thorough, let's check the other options too.Option C: ( (a + b)^2 = a^2 + b^2 ). I remember that when you square a binomial like ( (a + b) ), you have to use the formula ( (a + b)^2 = a^2 + 2ab + b^2 ). So, expanding ( (a + b)^2 ) should give me ( a^2 + 2ab + b^2 ), not just ( a^2 + b^2 ). The middle term ( 2ab ) is missing here, which means option C is incorrect.Lastly, option D: ( sqrt{2} times sqrt{6} = sqrt{3} ). To multiply square roots, you can combine them under a single square root. So, ( sqrt{2} times sqrt{6} = sqrt{2 times 6} = sqrt{12} ). Now, ( sqrt{12} ) can be simplified by factoring out perfect squares. ( 12 = 4 times 3 ), and ( sqrt{4} = 2 ), so ( sqrt{12} = 2sqrt{3} ). But the right side of the equation is ( sqrt{3} ), not ( 2sqrt{3} ). Therefore, option D is also incorrect.So, after going through all four options, only option B seems to be correct.

💡Okay, so I need to find the equation of a line that passes through the point P(2, -3) and also passes through the intersection point of two other lines: 3x + 2y - 4 = 0 and x - y + 5 = 0. Hmm, let me break this down step by step.First, I think I need to find where these two lines intersect. That means solving the system of equations:3x + 2y - 4 = 0 x - y + 5 = 0Alright, so I have two equations with two variables. Maybe I can solve one of them for one variable and substitute into the other. Let me try solving the second equation for x.From the second equation: x - y + 5 = 0 So, x = y - 5Now, substitute this expression for x into the first equation:3(y - 5) + 2y - 4 = 0 Let me expand that: 3y - 15 + 2y - 4 = 0 Combine like terms: 5y - 19 = 0 So, 5y = 19 Therefore, y = 19/5Now that I have y, I can find x using x = y - 5: x = (19/5) - 5 Convert 5 to fifths: 5 = 25/5 So, x = (19/5) - (25/5) = (-6/5)Okay, so the intersection point of the two lines is (-6/5, 19/5). Got that.Now, I need to find the equation of the line that passes through P(2, -3) and (-6/5, 19/5). To find the equation of a line, I can use the two-point form. The formula is:(y - y1) = m(x - x1)Where m is the slope, and (x1, y1) is a point on the line. I can use either of the two points to plug into this formula. First, I need to find the slope m.The slope m between two points (x1, y1) and (x2, y2) is given by:m = (y2 - y1)/(x2 - x1)Let me label the points: Point A: P(2, -3) Point B: (-6/5, 19/5)So, plugging into the slope formula:m = (19/5 - (-3)) / (-6/5 - 2)First, simplify the numerator: 19/5 - (-3) = 19/5 + 3 = 19/5 + 15/5 = 34/5Denominator: -6/5 - 2 = -6/5 - 10/5 = -16/5So, m = (34/5) / (-16/5) = (34/5) * (-5/16) = -34/16 = -17/8Okay, so the slope m is -17/8.Now, using the point-slope form with point P(2, -3):y - (-3) = (-17/8)(x - 2) Simplify: y + 3 = (-17/8)(x - 2)Now, let's convert this into standard form (Ax + By + C = 0). First, distribute the slope:y + 3 = (-17/8)x + (17/8)*2 Calculate (17/8)*2 = 34/8 = 17/4So, y + 3 = (-17/8)x + 17/4Now, let's get all terms to one side:(17/8)x + y + 3 - 17/4 = 0To combine the constants, convert 3 to quarters: 3 = 12/4 So, 12/4 - 17/4 = -5/4Thus, the equation becomes:(17/8)x + y - 5/4 = 0To eliminate the fractions, multiply every term by 8 (the least common denominator):17x + 8y - 10 = 0So, the equation of the line is 17x + 8y - 10 = 0.Wait, let me double-check my calculations to make sure I didn't make a mistake.Starting from the slope: m = (19/5 - (-3)) / (-6/5 - 2) = (19/5 + 15/5) / (-6/5 - 10/5) = (34/5) / (-16/5) = -34/16 = -17/8 That seems correct.Point-slope form: y + 3 = (-17/8)(x - 2) Distribute: y + 3 = (-17/8)x + 34/8 = (-17/8)x + 17/4 Subtract (-17/8)x and add 17/4 to both sides: (17/8)x + y + 3 - 17/4 = 0 Convert 3 to 12/4: (17/8)x + y + 12/4 - 17/4 = 0 = (17/8)x + y - 5/4 = 0 Multiply by 8: 17x + 8y - 10 = 0 Yes, that looks right.Alternatively, I could have used the other point (-6/5, 19/5) to write the equation, but I think the result should be the same.Let me verify by plugging in both points into the final equation.First, point P(2, -3): 17(2) + 8(-3) - 10 = 34 - 24 - 10 = 0 Yes, that works.Second, point (-6/5, 19/5): 17*(-6/5) + 8*(19/5) - 10 = (-102/5) + (152/5) - 10 = (50/5) - 10 = 10 - 10 = 0 That also works.Okay, so both points satisfy the equation. I think I did it correctly.