💡Okay, so I have this problem where I need to find the value of x + y given that the determinant of a certain 3x3 matrix is zero. The matrix is:[begin{vmatrix} 2 & 5 & 10 4 & x & y 4 & y & x end{vmatrix} = 0]Hmm, determinants. I remember that for a 3x3 matrix, the determinant can be calculated by expanding along a row or column using minors. Maybe I should expand along the first row since it has some smaller numbers, which might make the calculations easier.So, the determinant expansion along the first row would be:2 times the determinant of the submatrix obtained by removing the first row and first column, minus 5 times the determinant of the submatrix obtained by removing the first row and second column, plus 10 times the determinant of the submatrix obtained by removing the first row and third column.Let me write that out step by step.First, the submatrix for the element 2 (which is in position (1,1)) is:[begin{vmatrix} x & y y & x end{vmatrix}]The determinant of this 2x2 matrix is (x)(x) - (y)(y) = x² - y².Next, the submatrix for the element 5 (which is in position (1,2)) is:[begin{vmatrix} 4 & y 4 & x end{vmatrix}]The determinant here is (4)(x) - (4)(y) = 4x - 4y.Then, the submatrix for the element 10 (which is in position (1,3)) is:[begin{vmatrix} 4 & x 4 & y end{vmatrix}]The determinant of this one is (4)(y) - (4)(x) = 4y - 4x.Putting it all together, the determinant of the original matrix is:2*(x² - y²) - 5*(4x - 4y) + 10*(4y - 4x)Let me compute each term separately.First term: 2*(x² - y²) = 2x² - 2y²Second term: -5*(4x - 4y) = -20x + 20yThird term: 10*(4y - 4x) = 40y - 40xNow, adding all these together:2x² - 2y² - 20x + 20y + 40y - 40xLet me combine like terms.For the x² and y² terms: 2x² - 2y²For the x terms: -20x - 40x = -60xFor the y terms: 20y + 40y = 60ySo, the determinant simplifies to:2x² - 2y² - 60x + 60yHmm, I notice that 2x² - 2y² can be factored as 2(x² - y²), which is 2(x - y)(x + y). Also, the other terms can be factored as -60x + 60y = -60(x - y). So, let's factor out (x - y) from the entire expression.So, 2(x - y)(x + y) - 60(x - y) = (x - y)(2(x + y) - 60)Factor out the 2 from the second term inside the parentheses:= (x - y)(2(x + y - 30))So, the determinant is equal to 2(x - y)(x + y - 30).But the problem states that this determinant is zero. So, 2(x - y)(x + y - 30) = 0.Now, since 2 is not zero, we can divide both sides by 2, giving:(x - y)(x + y - 30) = 0Now, for a product of two terms to be zero, at least one of the terms must be zero. So, either x - y = 0 or x + y - 30 = 0.But wait, the problem says that x and y are distinct real numbers. So, x - y ≠ 0 because x ≠ y. Therefore, the other factor must be zero.So, x + y - 30 = 0, which implies x + y = 30.Therefore, the value of x + y is 30.I think that's the answer. Let me just double-check my steps to make sure I didn't make any mistakes.First, I expanded the determinant correctly along the first row, computing each minor. Then, I expanded each minor correctly, subtracting the products. Then, I combined like terms correctly, factoring out common terms. I noticed that (x - y) was a common factor, which is valid because x ≠ y, so we can safely divide by (x - y). Thus, the remaining factor gives x + y = 30.Yeah, that seems solid. I don't see any errors in my calculations.**Final Answer**The value of ( x + y ) is boxed{30}.

💡Okay, so I have this problem here where I need to find the value of 'a' for the function f(x) = log base 1/2 of (a/(x² + 1)). The function is defined on all real numbers, and its range is given as [-1, +∞). The options are A: 1/4, B: 1/2, C: 1, D: 2. Hmm, let's try to figure this out step by step.First, I remember that the logarithm function log_b(y) has different behaviors depending on whether the base b is greater than 1 or between 0 and 1. In this case, the base is 1/2, which is less than 1. So, the logarithm function log_{1/2}(y) is a decreasing function. That means as y increases, the value of the logarithm decreases, and vice versa.Now, the function f(x) is given as log_{1/2}(a/(x² + 1)). Let me denote the argument of the logarithm as u, so u = a/(x² + 1). Then, f(x) = log_{1/2}(u). Since log_{1/2} is a decreasing function, the behavior of f(x) will depend on the behavior of u.I need to find the range of f(x), which is given as [-1, +∞). So, f(x) can take any value from -1 up to infinity. Since the logarithm function can take any real value, but in this case, it's restricted to start at -1 and go upwards. Let me think about how this relates to the argument u.Since log_{1/2}(u) is decreasing, the maximum value of f(x) occurs when u is minimized, and the minimum value of f(x) occurs when u is maximized. Wait, but the range is [-1, +∞), so the smallest value f(x) can take is -1, and it goes up to infinity. That means that the maximum value of f(x) is infinity, and the minimum is -1.But wait, actually, since log_{1/2}(u) is decreasing, when u approaches 0 from the right, log_{1/2}(u) approaches infinity, and when u approaches infinity, log_{1/2}(u) approaches negative infinity. But in our case, the range is [-1, +∞), which suggests that the function f(x) can go up to infinity but only as low as -1. So, that means that u cannot go beyond a certain point where log_{1/2}(u) would be less than -1.So, let's formalize this. The range of f(x) is [-1, +∞). Therefore, the minimum value of f(x) is -1, which corresponds to the maximum value of u. Because log_{1/2}(u) is decreasing, the maximum u will give the minimum f(x). So, let's set f(x) = -1 and solve for u.So, log_{1/2}(u) = -1. To solve for u, I can rewrite this equation in exponential form. Remember that log_b(a) = c is equivalent to b^c = a. So, (1/2)^{-1} = u. Calculating that, (1/2)^{-1} is equal to 2. So, u = 2.Therefore, the maximum value of u is 2. Since u = a/(x² + 1), the maximum value of u occurs when the denominator x² + 1 is minimized. The expression x² + 1 is always greater than or equal to 1 because x² is non-negative. The minimum value of x² + 1 is 1, which occurs when x = 0.So, substituting x = 0 into u, we get u = a/(0 + 1) = a. Therefore, the maximum value of u is a. But we found earlier that the maximum value of u is 2. Therefore, a must be equal to 2.Let me double-check this. If a = 2, then u = 2/(x² + 1). The maximum value of u is 2 when x = 0, and as x moves away from 0, u decreases towards 0. Since log_{1/2}(u) is decreasing, when u is 2, log_{1/2}(2) = -1, and as u approaches 0, log_{1/2}(u) approaches infinity. So, the range of f(x) is indeed [-1, +∞), which matches the given range.Therefore, the value of 'a' is 2, which corresponds to option D.**Final Answer**boxed{D}

💡Okay, so I need to find the equation of a circle that passes through point A(4,1) and is tangent to the line x - y - 1 = 0 at point B(2,1). Hmm, let me think about how to approach this.First, I remember that the general equation of a circle is (x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center and r is the radius. So, I need to find a, b, and r.Since the circle passes through point A(4,1), plugging that into the circle's equation gives me:(4 - a)^2 + (1 - b)^2 = r^2. Let me write that down as equation (1).Also, the circle is tangent to the line x - y - 1 = 0 at point B(2,1). That means point B is on the circle, so plugging B into the circle's equation gives:(2 - a)^2 + (1 - b)^2 = r^2. Let me call this equation (2).Now, if I subtract equation (2) from equation (1), I can eliminate r^2 and get an equation involving a and b.So, (4 - a)^2 - (2 - a)^2 = 0.Let me expand both squares:(16 - 8a + a^2) - (4 - 4a + a^2) = 0Simplify:16 - 8a + a^2 - 4 + 4a - a^2 = 0Combine like terms:(16 - 4) + (-8a + 4a) + (a^2 - a^2) = 012 - 4a = 0So, 12 = 4a => a = 3.Alright, so the x-coordinate of the center is 3. Now, I need to find b.Since the circle is tangent to the line x - y - 1 = 0 at point B(2,1), the radius at the point of tangency should be perpendicular to the tangent line. The slope of the tangent line is given by the coefficient of y in the line equation. Let me rewrite the line equation in slope-intercept form:x - y - 1 = 0 => y = x - 1.So, the slope of the tangent line is 1. Therefore, the slope of the radius at point B should be the negative reciprocal, which is -1.The radius from the center (a, b) to point B(2,1) has a slope of (1 - b)/(2 - a). Since we found a = 3, this becomes (1 - b)/(2 - 3) = (1 - b)/(-1) = b - 1.We know this slope should be -1, so:b - 1 = -1 => b = 0.Great, so the center of the circle is at (3, 0). Now, let's find the radius r using equation (2):(2 - 3)^2 + (1 - 0)^2 = r^2(-1)^2 + (1)^2 = r^21 + 1 = r^2 => r^2 = 2 => r = √2.Wait, hold on, that doesn't seem right. Let me double-check. If the center is (3,0), then the distance from (3,0) to (2,1) is sqrt[(3-2)^2 + (0-1)^2] = sqrt[1 + 1] = sqrt(2). So, r is sqrt(2). But let me check equation (1):(4 - 3)^2 + (1 - 0)^2 = 1 + 1 = 2, which is equal to r^2. So, that's consistent.Wait, but in my initial calculation, I thought r was 2, but that must have been a mistake. So, actually, r is sqrt(2). Hmm, but let me think again.Wait, no, hold on. If the center is (3,0), then the distance from (3,0) to (4,1) is sqrt[(4-3)^2 + (1-0)^2] = sqrt[1 + 1] = sqrt(2), which is the same as the radius. So, that's correct.But in my initial thought process, I thought r was 2, which was incorrect. So, the radius is sqrt(2). Therefore, the equation of the circle is (x - 3)^2 + y^2 = 2.Wait, but let me make sure. The problem says the circle is tangent to the line x - y - 1 = 0 at B(2,1). So, the distance from the center (3,0) to the line should be equal to the radius.The formula for the distance from a point (a,b) to the line Ax + By + C = 0 is |A*a + B*b + C| / sqrt(A^2 + B^2).So, for the line x - y - 1 = 0, A=1, B=-1, C=-1.Distance = |1*3 + (-1)*0 -1| / sqrt(1 + 1) = |3 - 0 -1| / sqrt(2) = |2| / sqrt(2) = 2/sqrt(2) = sqrt(2).Which is equal to the radius. So, that checks out.Therefore, the equation of the circle is (x - 3)^2 + y^2 = 2.Wait, but in my initial thought, I thought r was 2, but that was a mistake. So, the correct radius is sqrt(2), and the equation is (x - 3)^2 + y^2 = 2.But let me check again. If I plug in point A(4,1):(4 - 3)^2 + (1)^2 = 1 + 1 = 2, which is correct.And point B(2,1):(2 - 3)^2 + (1)^2 = 1 + 1 = 2, which is also correct.And the distance from center (3,0) to the line is sqrt(2), which is equal to the radius, so it's tangent.Therefore, the equation is (x - 3)^2 + y^2 = 2.Wait, but in the initial problem, the answer was given as (x - 3)^2 + y^2 = 4, which would mean r = 2. But according to my calculations, r is sqrt(2). So, I must have made a mistake somewhere.Let me go back. When I subtracted equation (2) from equation (1), I got a = 3, which seems correct. Then, for the slope, I said the slope of the radius is (1 - b)/(2 - a) = (1 - b)/(-1) = b - 1. And since the slope should be -1, b - 1 = -1 => b = 0. That seems correct.Then, plugging into equation (2): (2 - 3)^2 + (1 - 0)^2 = 1 + 1 = 2 = r^2. So, r = sqrt(2). So, the equation is (x - 3)^2 + y^2 = 2.But the initial answer was (x - 3)^2 + y^2 = 4. So, perhaps I made a mistake in the slope calculation.Wait, the slope of the tangent line is 1, so the slope of the radius should be -1. The slope from center (3,0) to point B(2,1) is (1 - 0)/(2 - 3) = 1/(-1) = -1, which is correct. So, that part is fine.Wait, but if the radius is sqrt(2), then the distance from center to the line is also sqrt(2), which is correct for tangency.So, perhaps the initial answer was wrong. Let me check again.Wait, if I consider the circle (x - 3)^2 + y^2 = 4, then the radius is 2. Let's see if that works.Distance from center (3,0) to line x - y -1 = 0 is |3 - 0 -1| / sqrt(1 + 1) = 2 / sqrt(2) = sqrt(2). But the radius is 2, which is larger than sqrt(2), so the circle would intersect the line at two points, not just tangent. Therefore, the initial answer must be wrong.So, the correct equation is (x - 3)^2 + y^2 = 2.Wait, but in the initial problem, the user wrote:"A circle C passes through point A(4,1) and is tangent to the line x - y -1=0 at point B(2,1). Find the equation of circle C."And the initial answer was (x - 3)^2 + y^2 = 4, which is incorrect. So, I think my corrected answer is (x - 3)^2 + y^2 = 2.But let me double-check everything again.Center at (3,0). Distance to A(4,1): sqrt[(4-3)^2 + (1-0)^2] = sqrt(1 + 1) = sqrt(2). Distance to B(2,1): same. Distance to line: sqrt(2). So, radius is sqrt(2). Therefore, equation is (x - 3)^2 + y^2 = 2.Yes, that seems correct. So, the initial answer was wrong, and the correct equation is (x - 3)^2 + y^2 = 2.

💡Alright, so I've got this geometry problem here about skew lines in space. It says, "Given two skew lines in space, the relationship between two lines that are respectively parallel to these skew lines is..." and then it gives four options: Always parallel, Always skew, Intersect or skew, or Always intersect. Hmm, okay, let me try to wrap my head around this.First off, I need to remember what skew lines are. From what I recall, skew lines are lines in three-dimensional space that are neither parallel nor intersecting. So, they just kind of go off in different directions without ever meeting. That makes sense. So, if I have two skew lines, they don't lie on the same plane, right? They're not parallel, and they don't cross each other.Now, the question is about lines that are respectively parallel to these skew lines. So, if I have two skew lines, let's call them Line A and Line B, and then I have two other lines, Line C and Line D, where Line C is parallel to Line A, and Line D is parallel to Line B. The question is asking about the relationship between Line C and Line D.Okay, so I need to figure out if Line C and Line D are always parallel, always skew, can intersect or be skew, or always intersect. Let's think about this step by step.First, let's consider the properties of parallel lines. If two lines are parallel, they never intersect, and they maintain a constant distance from each other. So, if Line C is parallel to Line A, and Line D is parallel to Line B, what does that mean for Line C and Line D?Well, since Line A and Line B are skew, they aren't parallel, and they don't intersect. Now, if Line C is parallel to Line A, and Line D is parallel to Line B, could Line C and Line D be parallel? Hmm, if Line C were parallel to Line D, that would mean Line A is parallel to Line B, right? Because parallelism is transitive. But wait, Line A and Line B are skew, so they aren't parallel. Therefore, Line C and Line D can't be parallel either. Okay, so that rules out option A: Always parallel.Next, could Line C and Line D always be skew? Well, skew lines are non-parallel, non-intersecting lines. So, if Line C and Line D are not parallel, they could either be skew or intersecting. But the question is asking about their relationship. So, it's possible that Line C and Line D are skew, but is that always the case?Wait, let me think. If I translate Line A to get Line C, and translate Line B to get Line D, depending on how I translate them, Line C and Line D could either be skew or intersecting. For example, if I translate Line A and Line B in such a way that their translations end up on the same plane, they might intersect. But if I translate them in different directions, they might remain skew.So, it seems like Line C and Line D could either be skew or intersecting, depending on the translations. That means they aren't always skew, so option B is out.What about option D: Always intersect? Well, if Line C and Line D are not parallel, they could intersect or be skew. So, they don't always intersect. Therefore, option D is also incorrect.That leaves us with option C: Intersect or skew. Yeah, that makes sense. Depending on how the original skew lines are positioned and how we translate them to get the parallel lines, the resulting lines could either intersect or remain skew.Let me double-check this reasoning. If two lines are parallel to skew lines, they can't be parallel themselves because that would imply the original lines were parallel, which they aren't. So, they have to be either intersecting or skew. It depends on their relative positions in space. If they're in the same plane after translation, they might intersect; otherwise, they stay skew.Yeah, I think that's right. So, the correct answer should be C: Intersect or skew.

💡First, I need to determine how much Mike earned from his second job by subtracting the amount he earned from his first job from his total earnings.Total earnings: 160Earnings from the first job: 52Earnings from the second job = Total earnings - Earnings from the first jobEarnings from the second job = 160 - 52Earnings from the second job = 108Next, I'll calculate the number of hours Mike worked at his second job by dividing his earnings from the second job by his hourly wage.Hourly wage: 9 per hourHours worked at the second job = Earnings from the second job / Hourly wageHours worked at the second job = 108 / 9Hours worked at the second job = 12Therefore, Mike worked 12 hours at his second job.

💡First, I need to determine Sandy's savings from last year and this year.Let’s denote Sandy's annual salary last year as ( S ).**Last Year's Savings:**Sandy saved 10% of her salary.[text{Last Year's Savings} = 0.10 times S]**This Year's Salary:**Sandy made 10% more money than last year.[text{This Year's Salary} = S + 0.10 times S = 1.10 times S]**This Year's Savings:**Sandy saved 6% of her new salary.[text{This Year's Savings} = 0.06 times (1.10 times S) = 0.066 times S]**Calculating the Percentage:**To find out what percent the amount saved this year is of the amount saved last year, I divide this year's savings by last year's savings and multiply by 100.[text{Percentage} = left( frac{0.066 times S}{0.10 times S} right) times 100 = left( frac{0.066}{0.10} right) times 100 = 66%]