💡Okay, so I have this problem here about the eccentricity of a conic section. The equation given is (frac{x^2}{m} + y^2 = 1), and the eccentricity is (sqrt{7}). I need to find the value of (m) from the options provided: A is (frac{1}{6}), B is 6, C is (-frac{1}{6}), and D is (-6). First, I remember that conic sections include ellipses, hyperbolas, parabolas, etc. The general form of an ellipse is (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), and for a hyperbola, it's (frac{x^2}{a^2} - frac{y^2}{b^2} = 1) or (frac{y^2}{b^2} - frac{x^2}{a^2} = 1). Looking at the given equation, (frac{x^2}{m} + y^2 = 1), it seems similar to an ellipse because both terms are positive. However, I also recall that if the coefficients of (x^2) and (y^2) are of opposite signs, it's a hyperbola. So, I need to figure out whether this is an ellipse or a hyperbola.Wait, in the given equation, both terms are positive because (y^2) is the same as (frac{y^2}{1}), so it's (frac{x^2}{m} + frac{y^2}{1} = 1). For this to be an ellipse, both (m) and 1 should be positive. But if (m) is negative, then the equation would actually represent a hyperbola because one term would effectively be negative.So, let me check the sign of (m). If (m) is positive, it's an ellipse; if (m) is negative, it's a hyperbola. Since the eccentricity is given as (sqrt{7}), which is greater than 1, that tells me it's a hyperbola because the eccentricity of an ellipse is less than 1, and for a hyperbola, it's greater than 1.Okay, so it must be a hyperbola. Therefore, (m) must be negative. That narrows down the options to C and D: (-frac{1}{6}) and (-6).Now, let's write the equation in the standard form of a hyperbola. Since the positive term is with (y^2), the hyperbola opens up and down. The standard form is (frac{y^2}{a^2} - frac{x^2}{b^2} = 1). Comparing this with our equation, (frac{x^2}{m} + y^2 = 1), I can rewrite it as (frac{y^2}{1} - frac{x^2}{-m} = 1). So, (a^2 = 1) and (b^2 = -m). For hyperbolas, the relationship between (a), (b), and (c) (the distance from the center to the foci) is (c^2 = a^2 + b^2). The eccentricity (e) is defined as (e = frac{c}{a}). Given that (e = sqrt{7}), we can write:[e = frac{c}{a} = sqrt{7}]Since (a^2 = 1), (a = 1). Therefore:[c = a cdot e = 1 cdot sqrt{7} = sqrt{7}]Now, using the relationship (c^2 = a^2 + b^2):[(sqrt{7})^2 = 1^2 + b^2][7 = 1 + b^2][b^2 = 7 - 1 = 6]But earlier, we found that (b^2 = -m). So:[-m = 6][m = -6]Therefore, the value of (m) is (-6), which corresponds to option D.Let me double-check my steps to make sure I didn't make a mistake. 1. Identified that the equation could be an ellipse or hyperbola based on the sign of (m).2. Noted that eccentricity greater than 1 implies a hyperbola, so (m) must be negative.3. Rewrote the equation in standard hyperbola form, identifying (a^2 = 1) and (b^2 = -m).4. Used the eccentricity formula (e = c/a) to find (c = sqrt{7}).5. Applied the hyperbola relationship (c^2 = a^2 + b^2) to solve for (b^2 = 6).6. Concluded that (m = -6).Everything seems to check out. I don't see any errors in my reasoning. So, I'm confident that the correct answer is D: (-6).**Final Answer**boxed{D}

💡Okay, so I have this quadratic function, right? It's given by the equation y equals a x squared plus b x plus c. The question is asking about the number of zeros this function has, given that the product of a and c is less than zero. Hmm, okay, so ac is negative. I need to figure out how that affects the number of zeros.First, I remember that the number of zeros of a quadratic function is determined by its discriminant. The discriminant formula is b squared minus four a c. If the discriminant is positive, there are two real zeros; if it's zero, there's exactly one real zero; and if it's negative, there are no real zeros, meaning the function doesn't cross the x-axis.So, in this case, since ac is less than zero, that means a and c have opposite signs. One of them is positive, and the other is negative. Let me think about how that affects the discriminant. The discriminant is b squared minus four a c. If a and c are of opposite signs, then their product ac is negative. So, four a c would be negative as well because four is positive, and a c is negative. Therefore, subtracting a negative number is like adding a positive number. So, the discriminant becomes b squared plus a positive number. Wait, so that means the discriminant is definitely going to be positive because b squared is always non-negative, and we're adding another positive number to it. So, regardless of the value of b, the discriminant will be greater than zero. That means the quadratic equation will have two distinct real roots. Let me double-check that. If a is positive and c is negative, then ac is negative. Similarly, if a is negative and c is positive, ac is still negative. In both cases, four a c is negative, so when we subtract that from b squared, it's like adding a positive number. So, yes, the discriminant is definitely positive. Therefore, the quadratic function will cross the x-axis at two distinct points, meaning there are two zeros. I don't think there's any case where ac being negative would result in a non-positive discriminant because even if b is zero, b squared is zero, and then the discriminant would be negative four a c, which is positive since a c is negative. So, even in that case, the discriminant is positive. So, putting it all together, if ac is less than zero, the discriminant is positive, leading to two real zeros for the quadratic function. I think that makes sense. I don't see any flaws in this reasoning. Maybe I can think of a specific example to test it out.Let's say a is 1 and c is -1, so ac is -1, which is less than zero. Let's choose b as 0 for simplicity. Then the quadratic equation becomes y equals x squared minus 1. The discriminant here is 0 squared minus four times one times (-1), which is 0 plus 4, so 4. That's positive, and indeed, the equation x squared minus 1 equals zero has two real roots: x equals 1 and x equals -1.Another example: a is -2, c is 3, so ac is -6, which is less than zero. Let's pick b as 4. The quadratic equation is y equals -2x squared plus 4x plus 3. The discriminant is 4 squared minus four times (-2) times 3, which is 16 plus 24, so 40. That's positive, so again, two real roots. Wait, let me try with b being zero again. If a is -1 and c is 2, then ac is -2. The equation is y equals -x squared plus 2. The discriminant is 0 minus four times (-1) times 2, which is 0 plus 8, so 8. Positive again, two real roots. What if b is a large number? Let's say a is 1, c is -1, and b is 10. The equation is y equals x squared plus 10x minus 1. The discriminant is 100 minus four times one times (-1), which is 100 plus 4, so 104. Still positive. So, regardless of the value of b, as long as ac is negative, the discriminant is positive.Is there any case where ac is negative, but the discriminant is zero or negative? Let me see. Suppose a is 1, c is -1, and b is 2. The discriminant is 4 minus four times one times (-1), which is 4 plus 4, so 8. Still positive. If b is 1, discriminant is 1 plus 4, which is 5. If b is 0, discriminant is 0 plus 4, which is 4. If b is negative, say b is -3, discriminant is 9 plus 4, which is 13. So, no matter what b is, as long as ac is negative, discriminant is positive.Therefore, I can confidently say that if ac is less than zero, the quadratic function will have two real zeros. So, the answer is two.**Final Answer**The number of zeros of the function is boxed{2}.

💡First, I need to determine the amount of water and solids in the 1.6 kg of flower-nectar. Since the nectar contains 50% water, there are 0.8 kg of water and 0.8 kg of solids.Next, I recognize that the bees process the nectar to produce 1 kg of honey. During this process, the solids remain unchanged, so the 0.8 kg of solids become part of the honey.To find out how much water is removed, I subtract the weight of the honey from the total weight of the nectar: 1.6 kg - 1 kg = 0.6 kg of water removed.Finally, I calculate the remaining water in the honey by subtracting the removed water from the initial water content: 0.8 kg - 0.6 kg = 0.2 kg of water remaining.To determine the percentage of water in the honey, I divide the remaining water by the total weight of the honey and multiply by 100: (0.2 kg / 1 kg) * 100 = 20%.Therefore, the honey contains 20% water.

💡Alright, so I've got this problem here where I need to find the probability that a certain quadratic function has a root. The function is given as ( f(x) = x^2 + 2ax - b^2 + pi ), and the numbers ( a ) and ( b ) are randomly selected from the interval ([-π, π]). The options given are A: ( frac{7}{8} ), B: ( frac{3}{4} ), C: ( frac{1}{2} ), and D: ( frac{1}{4} ).Okay, let's start by understanding what it means for a quadratic function to have a root. A quadratic equation ( ax^2 + bx + c = 0 ) has real roots if its discriminant is non-negative. The discriminant is given by ( D = b^2 - 4ac ). If ( D geq 0 ), the equation has real roots.In this problem, the function is ( f(x) = x^2 + 2ax - b^2 + pi ). Let me rewrite it in the standard quadratic form:( f(x) = x^2 + 2a x + (-b^2 + pi) )So, comparing to ( ax^2 + bx + c ), here:- ( a = 1 ) (coefficient of ( x^2 ))- ( b = 2a ) (coefficient of ( x ))- ( c = -b^2 + pi ) (constant term)Now, let's compute the discriminant ( D ):( D = (2a)^2 - 4 times 1 times (-b^2 + pi) )( D = 4a^2 - 4(-b^2 + pi) )( D = 4a^2 + 4b^2 - 4pi )For the quadratic to have real roots, we need ( D geq 0 ):( 4a^2 + 4b^2 - 4pi geq 0 )Divide both sides by 4:( a^2 + b^2 - pi geq 0 )So,( a^2 + b^2 geq pi )Alright, so the condition for the function to have a root is that ( a^2 + b^2 geq pi ).Now, since ( a ) and ( b ) are both selected randomly from the interval ([-π, π]), we can model this as a geometric probability problem. The sample space is all possible pairs ( (a, b) ) where ( a ) and ( b ) are in ([-π, π]). This forms a square in the coordinate system with side length ( 2π ), so the area of the sample space is ( (2π)^2 = 4π^2 ).The event we're interested in is ( a^2 + b^2 geq pi ). To find the probability, we need to find the area of the region where ( a^2 + b^2 geq pi ) within our square and then divide it by the total area of the square.Let me visualize this. The equation ( a^2 + b^2 = pi ) represents a circle with radius ( sqrt{pi} ) centered at the origin. So, the region ( a^2 + b^2 geq pi ) is the area outside this circle but still within the square ([-π, π] times [-π, π]).Therefore, the area we're interested in is the area of the square minus the area of the circle. The area of the circle is ( π times (sqrt{pi})^2 = π times π = π^2 ). So, the area where ( a^2 + b^2 geq pi ) is ( 4π^2 - π^2 = 3π^2 ).Wait, hold on. Is that correct? Because the circle with radius ( sqrt{pi} ) is entirely within the square since ( sqrt{pi} approx 1.772 ) and the square goes from (-π) to (π), which is approximately (-3.14) to (3.14). So, the circle is entirely inside the square, meaning the area outside the circle within the square is indeed ( 4π^2 - π^2 = 3π^2 ).Therefore, the probability ( P ) is:( P = frac{text{Area where } a^2 + b^2 geq pi}{text{Total area}} = frac{3π^2}{4π^2} = frac{3}{4} )So, the probability is ( frac{3}{4} ), which corresponds to option B.But wait, let me double-check my reasoning. Sometimes, when dealing with geometric probabilities, especially with circles inside squares, the area calculations can be tricky, especially if the circle extends beyond the square. However, in this case, since ( sqrt{pi} ) is approximately 1.772, which is less than ( π ) (approximately 3.14), the circle is entirely within the square. Therefore, subtracting the area of the circle from the square gives the correct area where ( a^2 + b^2 geq pi ).Another way to think about it is that the region ( a^2 + b^2 geq pi ) is the complement of the circle inside the square. So, the area we're interested in is indeed the total area minus the area of the circle.Therefore, my calculation seems correct. The probability is ( frac{3}{4} ), so the answer should be option B.But just to be thorough, let me consider if there's any overlap or edge cases where the circle might intersect the square. Since the radius of the circle is ( sqrt{pi} approx 1.772 ), and the square extends from (-π) to (π), which is about (-3.14) to (3.14), the circle doesn't reach the edges of the square. Therefore, there's no overlap issue, and the area calculation remains straightforward.So, I'm confident that the probability is ( frac{3}{4} ), which is option B.**Final Answer**boxed{B}

💡To determine the value of ( b ), we start by comparing the given equation ( x - 2y + b = 0 ) with the line equation ( y = frac{1}{2}x + b - 1 ).First, we rewrite the line equation in a form similar to the given equation:[y = frac{1}{2}x + b - 1]Multiplying both sides by 2 to eliminate the fraction:[2y = x + 2b - 2]Rearranging the terms:[x - 2y + 2b - 2 = 0]Now, we compare this with the given equation ( x - 2y + b = 0 ). By equating the constants, we find:[2b - 2 = b]Solving for ( b ):[2b - 2 = b implies b = 2]Thus, the value of ( b ) is 2.

💡First, I need to determine what the time was 2,880,717 minutes ago from the current time of 18 hours and 27 minutes. To do this, I'll start by converting the total minutes into hours and remaining minutes.Dividing 2,880,717 by 60 gives me 48,011 hours and 57 minutes. This means that 2,880,717 minutes is equivalent to 48,011 hours and 57 minutes.Next, I'll break down the 48,011 hours into more manageable chunks. I notice that 48,011 hours can be decomposed into 48,000 hours and 11 hours. Since there are 24 hours in a day, 48,000 hours is equal to 2,000 days. Therefore, 48,011 hours is equivalent to 2,000 days and 11 hours.Since days do not affect the clock time we're interested in, I'll focus on the remaining 11 hours and 57 minutes.Now, I'll adjust the current time of 18 hours and 27 minutes by subtracting the calculated 11 hours and 57 minutes. To make the subtraction easier, I'll break down the current time into 17 hours and 87 minutes.Performing the subtraction:- Subtract 11 hours from 17 hours, which gives 6 hours.- Subtract 57 minutes from 87 minutes, which gives 30 minutes.Therefore, the time 2,880,717 minutes earlier than 18:27 is 6 hours and 30 minutes.