💡First, I need to understand the relationships between the variables. We have x = (1/5)a and z = (1/5)b, where a and b are single-digit positive integers. This means that x and z can only take specific values based on the possible values of a and b.Next, I'll list out the possible values for a and b, which are 1 through 9. Then, I'll calculate the corresponding values for x and z by dividing each by 5. This gives me potential values for x and z such as 0.2, 0.4, 0.6, 0.8, 1.2, 1.4, 1.6, 1.8, and 2.Now, I need to consider the operation involving x and z that could not equal 33. I'll explore different operations like addition, subtraction, multiplication, and division. For addition and multiplication, the maximum possible results are 4 and 4, respectively, which are much smaller than 33. Therefore, these operations cannot result in 33.For subtraction, the maximum difference between x and z is 2, which is also much smaller than 33.For division, the result will be a fraction or a number less than or equal to 1, which again cannot be 33.Based on this analysis, the operation involving x and z that could not equal 33 is either subtraction or division.

💡Okay, so I have this problem here asking which condition cannot determine triangle ABC to be a right triangle. The options are A, B, C, and D. Let me go through each one step by step.Starting with option A: The angles are in the ratio 3:4:5. Hmm, I remember that in a triangle, the sum of all angles is 180 degrees. So if the angles are in the ratio 3:4:5, I can find each angle by dividing 180 into 12 parts (since 3+4+5=12). That means each part is 15 degrees. So angle A is 3*15=45 degrees, angle B is 4*15=60 degrees, and angle C is 5*15=75 degrees. None of these angles are 90 degrees, so this triangle is definitely not a right triangle. So option A cannot determine it to be a right triangle.Moving on to option B: It says angle A minus angle B equals angle C. Let me write that down: angle A - angle B = angle C. Also, I know that in any triangle, the sum of the angles is 180 degrees. So angle A + angle B + angle C = 180. If I substitute angle C from the first equation into the second, I get angle A + angle B + (angle A - angle B) = 180. Simplifying that, angle A + angle B + angle A - angle B = 2*angle A = 180. So angle A is 90 degrees. That means the triangle has a right angle at A, so it is a right triangle. So option B can determine it to be a right triangle.Option C: The sides are in the ratio 1:2:sqrt(3). I recall the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Let me check if this ratio satisfies that. Let's assign the sides as follows: let's say AB = 1, BC = 2, and AC = sqrt(3). Wait, but in a triangle, the hypotenuse is the longest side. So if BC is 2, which is longer than sqrt(3) (approximately 1.732), then BC would be the hypotenuse. Let's check: AB^2 + AC^2 = 1^2 + (sqrt(3))^2 = 1 + 3 = 4. And BC^2 is 2^2 = 4. So yes, it satisfies the Pythagorean theorem. Therefore, this is a right triangle. So option C can determine it to be a right triangle.Option D: The sides are given as AB=0.7, BC=2.4, and AC=2.5. Again, I'll apply the Pythagorean theorem. Let's see if 0.7^2 + 2.4^2 equals 2.5^2. Calculating each: 0.7^2 is 0.49, 2.4^2 is 5.76, and 2.5^2 is 6.25. Adding 0.49 and 5.76 gives 6.25, which is equal to 2.5^2. So this also satisfies the Pythagorean theorem, meaning it's a right triangle. So option D can determine it to be a right triangle.So, reviewing all options, only option A does not result in a right triangle because all its angles are less than 90 degrees. The other options either result in a right angle through angle relationships or satisfy the Pythagorean theorem with their side lengths.

💡Okay, so I have this polynomial ( f(x) = x^n + a_1 x^{n-1} + cdots + a_n ) with real coefficients and real roots ( b_1, b_2, ldots, b_n ). I need to prove that for ( x geq max { b_1, b_2, ldots, b_n } ), the inequality[f(x+1) geq frac{-1}{frac{1}{x - b_1} + frac{1}{x - b_2} + cdots + frac{1}{x - b_n}}]holds. Hmm, let me try to unpack this step by step.First, since ( f(x) ) is a polynomial of degree ( n ) with real roots, I can express it as[f(x) = (x - b_1)(x - b_2) cdots (x - b_n).]So, ( f(x+1) ) would be[f(x+1) = (x + 1 - b_1)(x + 1 - b_2) cdots (x + 1 - b_n).]Given that ( x geq max { b_1, b_2, ldots, b_n } ), each term ( x - b_i ) is non-negative, and hence each ( x + 1 - b_i ) is positive. Therefore, ( f(x+1) ) is positive because it's a product of positive terms.Now, looking at the right-hand side of the inequality, we have[frac{-1}{frac{1}{x - b_1} + frac{1}{x - b_2} + cdots + frac{1}{x - b_n}}.]Since ( x geq max { b_1, b_2, ldots, b_n } ), each denominator ( x - b_i ) is non-negative, so each term ( frac{1}{x - b_i} ) is positive or zero. However, since ( x ) is strictly greater than each ( b_i ) (because if ( x = max { b_i } ), then ( x - b_i ) for that particular ( i ) is zero, making ( frac{1}{x - b_i} ) undefined). So, I think the inequality is considering ( x > max { b_i } ), ensuring all denominators are positive.Therefore, the denominator in the right-hand side is a sum of positive terms, making the entire expression negative because of the negative sign in front. So, the right-hand side is negative, while the left-hand side ( f(x+1) ) is positive. Therefore, the inequality ( f(x+1) geq text{something negative} ) is always true because a positive number is always greater than a negative number.Wait, that seems too straightforward. Maybe I'm missing something here. Let me double-check.If ( x > max { b_i } ), then ( x - b_i > 0 ) for all ( i ), so ( frac{1}{x - b_i} > 0 ), and hence the sum ( frac{1}{x - b_1} + cdots + frac{1}{x - b_n} > 0 ). Therefore, the right-hand side is ( frac{-1}{text{positive}} = text{negative} ).Since ( f(x+1) ) is positive, as established earlier, then indeed ( f(x+1) geq text{negative number} ) is always true. So, is this inequality trivially true because one side is positive and the other is negative?But the problem statement specifies ( n geq 2 ), so maybe there's a more nuanced approach required here. Perhaps the problem is expecting a proof that isn't just based on the positivity of ( f(x+1) ) and negativity of the right-hand side.Alternatively, maybe the inequality is meant to be non-trivial, and I need to consider a different approach. Let me think again.Given that ( f(x) = (x - b_1)(x - b_2) cdots (x - b_n) ), then ( f(x+1) = (x + 1 - b_1)(x + 1 - b_2) cdots (x + 1 - b_n) ).Let me denote ( y = x - max { b_i } ). Since ( x geq max { b_i } ), ( y geq 0 ). Then, each ( x - b_i = y + (max { b_i } - b_i) geq y geq 0 ). So, each term ( x - b_i ) is at least ( y ), which is non-negative.But I'm not sure if this substitution helps. Maybe I should consider the reciprocal of the sum in the denominator.Let me denote ( S = frac{1}{x - b_1} + frac{1}{x - b_2} + cdots + frac{1}{x - b_n} ). So, the right-hand side is ( -1/S ).Therefore, the inequality is ( f(x+1) geq -1/S ). Since ( f(x+1) ) is positive and ( -1/S ) is negative, this inequality is always true. So, maybe the problem is just pointing out that ( f(x+1) ) is positive and the right-hand side is negative, hence the inequality holds.But perhaps the problem is expecting a more precise bound or a different interpretation. Maybe the inequality is meant to be in absolute value? Or perhaps the negative sign is a typo?Alternatively, maybe I need to consider the reciprocal of the sum in the denominator and relate it to ( f(x+1) ) in some other way.Wait, let me think about the relationship between ( f(x+1) ) and the sum ( S ). Since ( f(x+1) = prod_{i=1}^n (x + 1 - b_i) ), and ( S = sum_{i=1}^n frac{1}{x - b_i} ).Perhaps I can use the AM-GM inequality or some other inequality to relate these two expressions.Let me recall that for positive real numbers, the arithmetic mean is greater than or equal to the geometric mean. So, maybe I can apply that here.But ( f(x+1) ) is a product, and ( S ) is a sum. It's not straightforward to apply AM-GM directly. Alternatively, maybe I can consider the reciprocal of ( S ) and relate it to ( f(x+1) ).Alternatively, perhaps I can consider the reciprocal of ( f(x+1) ) and relate it to ( S ). Let me see.Wait, ( f(x+1) = prod_{i=1}^n (x + 1 - b_i) ). Let me denote ( t_i = x - b_i ), so ( t_i geq 0 ) since ( x geq b_i ). Then, ( x + 1 - b_i = t_i + 1 ). So, ( f(x+1) = prod_{i=1}^n (t_i + 1) ).Similarly, ( S = sum_{i=1}^n frac{1}{t_i} ).So, the inequality becomes:[prod_{i=1}^n (t_i + 1) geq frac{-1}{sum_{i=1}^n frac{1}{t_i}}.]But since ( t_i > 0 ), ( sum_{i=1}^n frac{1}{t_i} > 0 ), so the right-hand side is negative. Therefore, the inequality is ( prod_{i=1}^n (t_i + 1) geq text{negative number} ), which is always true because the left-hand side is positive.So, again, this seems trivial because a positive number is always greater than a negative number. Therefore, the inequality holds for all ( x geq max { b_i } ).But perhaps the problem is expecting a different approach or a more precise bound. Maybe I need to consider the reciprocal of the sum and relate it to the product in some non-trivial way.Alternatively, perhaps the problem is misstated, and the inequality is supposed to be in the opposite direction or without the negative sign. Let me check the original problem again.The problem states:"Prove that for ( x geq max { b_1, b_2, ldots, b_n } ),[f(x+1) geq frac{-1}{frac{1}{x - b_1} + frac{1}{x - b_2} + cdots + frac{1}{x - b_n}}.]"So, it's definitely as written. Therefore, perhaps the key is to recognize that the right-hand side is negative, and since ( f(x+1) ) is positive, the inequality is always true.But maybe the problem is expecting a proof that ( f(x+1) ) is bounded below by this negative number, which is trivial because ( f(x+1) ) is positive. So, perhaps the problem is just pointing out this triviality.Alternatively, maybe the problem is expecting a different interpretation or a more precise bound. Let me think again.Wait, perhaps the problem is expecting to show that ( f(x+1) ) is greater than or equal to the negative reciprocal of the sum, but in a non-trivial way, perhaps using some properties of polynomials or inequalities.Alternatively, maybe I need to consider the derivative of ( f(x) ) or some other property. Let me recall that ( f(x) ) has real roots, so its derivative ( f'(x) ) can be expressed in terms of the roots as well.But I'm not sure if that's directly helpful here. Let me think about the behavior of ( f(x+1) ) as ( x ) increases. Since ( f(x) ) is a polynomial of degree ( n ), ( f(x+1) ) will grow like ( x^n ) as ( x ) becomes large. On the other hand, the sum ( S = sum_{i=1}^n frac{1}{x - b_i} ) will behave like ( frac{n}{x} ) for large ( x ). Therefore, the right-hand side ( -1/S ) will behave like ( -x/n ), which tends to negative infinity as ( x ) increases.Meanwhile, ( f(x+1) ) tends to positive infinity as ( x ) increases. Therefore, for large ( x ), the inequality ( f(x+1) geq -1/S ) is certainly true because the left-hand side is growing positively and the right-hand side is going to negative infinity.But perhaps the problem is more about the behavior near ( x = max { b_i } ). Let me consider ( x ) just slightly larger than ( max { b_i } ). In this case, one of the terms ( x - b_i ) is very small, making ( S ) very large, so ( -1/S ) is very close to zero from the negative side. Meanwhile, ( f(x+1) ) is positive, so the inequality holds.Therefore, regardless of the value of ( x geq max { b_i } ), the inequality ( f(x+1) geq -1/S ) holds because the left-hand side is positive and the right-hand side is negative.But perhaps the problem is expecting a more formal proof rather than just an intuitive argument. Let me try to formalize this.Given that ( x geq max { b_1, b_2, ldots, b_n } ), we have ( x - b_i geq 0 ) for all ( i ). Therefore, ( x - b_i > 0 ) for all ( i ) (since if ( x = max { b_i } ), then ( x - b_i = 0 ) for that particular ( i ), making ( S ) undefined. So, we can assume ( x > max { b_i } )).Thus, ( f(x+1) = prod_{i=1}^n (x + 1 - b_i) ) is positive because each term ( x + 1 - b_i > 0 ).On the other hand, ( S = sum_{i=1}^n frac{1}{x - b_i} > 0 ) because each term ( frac{1}{x - b_i} > 0 ). Therefore, ( -1/S < 0 ).Since ( f(x+1) > 0 ) and ( -1/S < 0 ), it follows that ( f(x+1) geq -1/S ).Therefore, the inequality holds for all ( x > max { b_i } ).But maybe the problem is expecting a more involved proof, perhaps using inequalities like AM-GM or Cauchy-Schwarz. Let me explore that.Consider the AM-GM inequality, which states that for positive real numbers ( a_1, a_2, ldots, a_n ),[frac{a_1 + a_2 + cdots + a_n}{n} geq sqrt[n]{a_1 a_2 cdots a_n}.]Equality holds if and only if ( a_1 = a_2 = cdots = a_n ).In our case, we have ( f(x+1) = prod_{i=1}^n (x + 1 - b_i) ) and ( S = sum_{i=1}^n frac{1}{x - b_i} ).Let me denote ( t_i = x - b_i ), so ( t_i > 0 ) for all ( i ). Then, ( x + 1 - b_i = t_i + 1 ), and ( S = sum_{i=1}^n frac{1}{t_i} ).So, the inequality becomes:[prod_{i=1}^n (t_i + 1) geq frac{-1}{sum_{i=1}^n frac{1}{t_i}}.]But since ( prod_{i=1}^n (t_i + 1) > 0 ) and ( frac{-1}{sum_{i=1}^n frac{1}{t_i}} < 0 ), the inequality is trivially true.Alternatively, perhaps the problem is expecting us to consider the reciprocal of ( S ) and relate it to ( f(x+1) ) in a non-trivial way. Let me think about the relationship between ( f(x+1) ) and ( S ).Wait, perhaps I can consider the reciprocal of ( S ), which is ( frac{1}{S} = frac{1}{sum_{i=1}^n frac{1}{t_i}} ). This is the harmonic mean of the ( t_i )'s divided by ( n ). The harmonic mean is always less than or equal to the arithmetic mean, which is ( frac{sum t_i}{n} ).But I'm not sure how this relates directly to ( f(x+1) ).Alternatively, perhaps I can consider the reciprocal of ( f(x+1) ), which is ( frac{1}{prod_{i=1}^n (t_i + 1)} ). But I don't see a direct relationship between this and ( S ).Alternatively, maybe I can use the Cauchy-Schwarz inequality. The Cauchy-Schwarz inequality states that for positive real numbers ( a_i ) and ( b_i ),[left( sum_{i=1}^n a_i b_i right)^2 leq left( sum_{i=1}^n a_i^2 right) left( sum_{i=1}^n b_i^2 right).]But I'm not sure how to apply this here.Alternatively, perhaps I can consider the reciprocal of ( S ) and relate it to ( f(x+1) ) using some other inequality.Wait, let me consider the reciprocal of ( S ):[frac{1}{S} = frac{1}{sum_{i=1}^n frac{1}{t_i}}.]This is the harmonic mean of the ( t_i )'s divided by ( n ). The harmonic mean is always less than or equal to the geometric mean, which is ( sqrt[n]{prod_{i=1}^n t_i} ).So,[frac{n}{S} leq sqrt[n]{prod_{i=1}^n t_i}.]Therefore,[frac{1}{S} leq frac{1}{n} sqrt[n]{prod_{i=1}^n t_i}.]But ( prod_{i=1}^n t_i = prod_{i=1}^n (x - b_i) = f(x) ). So,[frac{1}{S} leq frac{1}{n} sqrt[n]{f(x)}.]But I'm not sure how this relates to ( f(x+1) ).Alternatively, perhaps I can consider the relationship between ( f(x+1) ) and ( f(x) ). Since ( f(x) = prod_{i=1}^n (x - b_i) ), then ( f(x+1) = prod_{i=1}^n (x + 1 - b_i) ).But I don't see a direct relationship that would help here.Alternatively, perhaps I can consider the derivative of ( f(x) ). The derivative ( f'(x) ) can be expressed as[f'(x) = sum_{i=1}^n prod_{j neq i} (x - b_j).]But I'm not sure if this helps with the given inequality.Alternatively, perhaps I can consider the reciprocal of ( f(x+1) ) and relate it to ( S ). Let me see:[frac{1}{f(x+1)} = frac{1}{prod_{i=1}^n (x + 1 - b_i)}.]But this doesn't seem directly related to ( S ).Alternatively, perhaps I can use the inequality between the arithmetic mean and the harmonic mean. The harmonic mean ( H ) of ( t_1, t_2, ldots, t_n ) is given by[H = frac{n}{sum_{i=1}^n frac{1}{t_i}} = frac{n}{S}.]And the arithmetic mean ( A ) is[A = frac{sum_{i=1}^n t_i}{n}.]We know that ( H leq A ), so[frac{n}{S} leq frac{sum_{i=1}^n t_i}{n}.]But again, I'm not sure how this relates to ( f(x+1) ).Alternatively, perhaps I can consider the reciprocal of ( S ) and relate it to ( f(x+1) ) using some other property.Wait, perhaps I can consider the reciprocal of ( S ) and relate it to the product ( f(x+1) ) using the inequality between the product and the sum.But I'm not sure.Alternatively, perhaps I can consider the reciprocal of ( S ) and relate it to ( f(x+1) ) using the Cauchy-Schwarz inequality in a different way.Let me think about the Cauchy-Schwarz inequality applied to the vectors ( (1, 1, ldots, 1) ) and ( left( frac{1}{sqrt{t_1}}, frac{1}{sqrt{t_2}}, ldots, frac{1}{sqrt{t_n}} right) ).Then,[left( sum_{i=1}^n 1 cdot frac{1}{sqrt{t_i}} right)^2 leq left( sum_{i=1}^n 1^2 right) left( sum_{i=1}^n frac{1}{t_i} right).]This simplifies to[left( sum_{i=1}^n frac{1}{sqrt{t_i}} right)^2 leq n cdot S.]But I don't see how this helps with the given inequality.Alternatively, perhaps I can consider the reciprocal of ( S ) and relate it to ( f(x+1) ) using the AM-GM inequality in a different way.Wait, let me consider the product ( f(x+1) cdot S ). That is,[f(x+1) cdot S = left( prod_{i=1}^n (t_i + 1) right) cdot left( sum_{i=1}^n frac{1}{t_i} right).]Perhaps I can find a lower bound for this product.Let me consider each term ( (t_i + 1) cdot frac{1}{t_i} = 1 + frac{1}{t_i} ). So,[f(x+1) cdot S = prod_{i=1}^n (t_i + 1) cdot sum_{i=1}^n frac{1}{t_i} = left( prod_{i=1}^n (t_i + 1) right) cdot left( sum_{i=1}^n frac{1}{t_i} right).]But I'm not sure how to proceed from here.Alternatively, perhaps I can use the inequality ( (1 + t_i) geq 2 sqrt{t_i} ) for each ( t_i > 0 ), which comes from the AM-GM inequality.So,[prod_{i=1}^n (1 + t_i) geq prod_{i=1}^n 2 sqrt{t_i} = 2^n sqrt{prod_{i=1}^n t_i}.]Therefore,[f(x+1) geq 2^n sqrt{prod_{i=1}^n t_i} = 2^n sqrt{f(x)}.]But I'm not sure how this helps with the given inequality.Alternatively, perhaps I can consider the reciprocal of ( S ) and relate it to ( f(x+1) ) using some other inequality.Wait, perhaps I can consider the reciprocal of ( S ) and relate it to the product ( f(x+1) ) using the inequality between the product and the sum.But I'm not making progress here. Let me try a different approach.Given that ( f(x+1) ) is positive and ( -1/S ) is negative, the inequality ( f(x+1) geq -1/S ) is always true. Therefore, the inequality holds for all ( x > max { b_i } ).But perhaps the problem is expecting a more formal proof, perhaps using induction or some other method.Alternatively, perhaps the problem is expecting to consider the case when ( n = 2 ) and then generalize.Let me try that. Let ( n = 2 ). Then, ( f(x) = (x - b_1)(x - b_2) ). So, ( f(x+1) = (x + 1 - b_1)(x + 1 - b_2) ).The sum ( S = frac{1}{x - b_1} + frac{1}{x - b_2} ).So, the inequality becomes:[(x + 1 - b_1)(x + 1 - b_2) geq frac{-1}{frac{1}{x - b_1} + frac{1}{x - b_2}}.]Let me compute the right-hand side:[frac{-1}{frac{1}{x - b_1} + frac{1}{x - b_2}} = frac{-1}{frac{(x - b_2) + (x - b_1)}{(x - b_1)(x - b_2)}} = frac{- (x - b_1)(x - b_2)}{2x - b_1 - b_2}.]So, the inequality becomes:[(x + 1 - b_1)(x + 1 - b_2) geq frac{ - (x - b_1)(x - b_2) }{2x - b_1 - b_2}.]Let me denote ( t_1 = x - b_1 ) and ( t_2 = x - b_2 ). Then, ( t_1, t_2 > 0 ), and the inequality becomes:[(t_1 + 1)(t_2 + 1) geq frac{ - t_1 t_2 }{2x - b_1 - b_2}.]But ( 2x - b_1 - b_2 = (x - b_1) + (x - b_2) = t_1 + t_2 ).So, the inequality becomes:[(t_1 + 1)(t_2 + 1) geq frac{ - t_1 t_2 }{t_1 + t_2}.]Multiplying both sides by ( t_1 + t_2 ) (which is positive), we get:[(t_1 + 1)(t_2 + 1)(t_1 + t_2) geq - t_1 t_2.]But the left-hand side is positive because all terms are positive, and the right-hand side is negative. Therefore, the inequality holds.So, for ( n = 2 ), the inequality holds because the left-hand side is positive and the right-hand side is negative.Similarly, for higher ( n ), the same reasoning applies: ( f(x+1) ) is positive, and the right-hand side is negative, so the inequality holds.Therefore, the inequality is trivially true because one side is positive and the other is negative.But perhaps the problem is expecting a more formal proof or a different approach. Let me try to write a formal proof.Given that ( f(x) = (x - b_1)(x - b_2) cdots (x - b_n) ) with real roots ( b_1, b_2, ldots, b_n ), and ( x geq max { b_1, b_2, ldots, b_n } ), we have ( x - b_i geq 0 ) for all ( i ). Since ( x ) is strictly greater than each ( b_i ) (to avoid division by zero), ( x - b_i > 0 ) for all ( i ).Therefore, ( f(x+1) = prod_{i=1}^n (x + 1 - b_i) ) is positive because each term ( x + 1 - b_i > 0 ).On the other hand, the sum ( S = sum_{i=1}^n frac{1}{x - b_i} ) is positive because each term ( frac{1}{x - b_i} > 0 ). Hence, ( -1/S ) is negative.Since ( f(x+1) > 0 ) and ( -1/S < 0 ), it follows that ( f(x+1) geq -1/S ).Therefore, the inequality holds for all ( x > max { b_i } ).In conclusion, the inequality is always true because the left-hand side is positive and the right-hand side is negative.

💡Alright, so I have this problem about an ellipse, and it's divided into two parts. Let me try to figure it out step by step.First, part (I) asks for the standard equation of the ellipse C. The ellipse is given by the equation x²/a² + y²/b² = 1, where a > b > 0. It passes through two points, P(1, √2/2) and Q(-√2, 0). So, I need to find the values of a² and b².Since the ellipse passes through point Q(-√2, 0), I can plug these coordinates into the ellipse equation. Let's do that:For point Q(-√2, 0):x = -√2, y = 0.Plugging into the ellipse equation:(-√2)² / a² + 0² / b² = 1Which simplifies to:(2) / a² = 1So, a² = 2. Therefore, a = √2.Now, we know a² is 2. Next, we can use point P(1, √2/2) to find b².For point P(1, √2/2):x = 1, y = √2/2.Plugging into the ellipse equation:1² / 2 + (√2/2)² / b² = 1Simplify:1/2 + ( (2/4) ) / b² = 1Which is:1/2 + (1/2) / b² = 1Let me write that as:1/2 + (1)/(2b²) = 1Subtract 1/2 from both sides:(1)/(2b²) = 1/2Multiply both sides by 2b²:1 = b²So, b² = 1, which means b = 1.Therefore, the standard equation of the ellipse is:x² / 2 + y² / 1 = 1Or, simplifying:x² / 2 + y² = 1Okay, that seems straightforward. So, part (I) is done.Now, part (II) is more complicated. It says: A line l passing through the right focus F of the ellipse intersects the ellipse C at points A and B, and intersects a circle with diameter FP at another point E (different from point F). We need to find the maximum value of |AB| · |FE|.First, let me recall some properties of ellipses. The standard form is x²/a² + y²/b² = 1, with a > b. The foci are located at (±c, 0), where c² = a² - b².Given that a² = 2 and b² = 1, c² = 2 - 1 = 1, so c = 1.Therefore, the foci are at (1, 0) and (-1, 0). The right focus F is at (1, 0).So, the line l passes through F(1, 0). Let me denote the line l. Since it's passing through (1, 0), I can write its equation in the form y = m(x - 1), where m is the slope. Alternatively, if I use a parameter t, I can write it as x = ty + 1, which might be useful for parametric equations.But let me think about the parametric form. Let me consider the line l in parametric terms. Suppose the line has a slope m, so its equation is y = m(x - 1). Alternatively, I can write it as x = ty + 1, where t is the parameter related to the slope.Wait, maybe using a parameter t for the slope is a good idea because it can help in parametrizing the points A and B.So, let me set the line l as x = ty + 1, where t is a parameter. Then, I can substitute this into the ellipse equation to find points A and B.Substituting x = ty + 1 into the ellipse equation x² / 2 + y² = 1:(ty + 1)² / 2 + y² = 1Expanding (ty + 1)²:= (t²y² + 2ty + 1) / 2 + y² = 1Multiply through:= (t²y²)/2 + ty + 1/2 + y² = 1Combine like terms:= (t²/2 + 1) y² + ty + (1/2 - 1) = 0= (t²/2 + 1) y² + ty - 1/2 = 0Multiply both sides by 2 to eliminate denominators:= (t² + 2) y² + 2ty - 1 = 0So, we have a quadratic equation in y:(t² + 2) y² + 2t y - 1 = 0Let me denote this as:A y² + B y + C = 0, where A = t² + 2, B = 2t, C = -1.The solutions for y are given by the quadratic formula:y = [-B ± √(B² - 4AC)] / (2A)Plugging in the values:y = [-2t ± √(4t² - 4*(t² + 2)*(-1))] / (2*(t² + 2))Simplify the discriminant:√(4t² - 4*(t² + 2)*(-1)) = √(4t² + 4(t² + 2)) = √(4t² + 4t² + 8) = √(8t² + 8) = √[8(t² + 1)] = 2√2√(t² + 1)So, the solutions are:y = [-2t ± 2√2√(t² + 1)] / [2(t² + 2)] = [-t ± √2√(t² + 1)] / (t² + 2)Therefore, the y-coordinates of points A and B are:y₁ = [-t + √2√(t² + 1)] / (t² + 2)y₂ = [-t - √2√(t² + 1)] / (t² + 2)Corresponding x-coordinates can be found using x = ty + 1.So, x₁ = t y₁ + 1x₂ = t y₂ + 1Now, we need to find |AB|. The distance between points A and B.Since both points lie on the line l, which has a slope m = t (since x = ty + 1 implies y = (x - 1)/t, so slope m = 1/t). Wait, actually, if x = ty + 1, then y = (x - 1)/t, so the slope is 1/t. Hmm, but I used t as a parameter, so maybe it's better to keep it as t for now.But for the distance |AB|, since they lie on a line with slope m, the distance can be expressed in terms of the difference in y-coordinates multiplied by √(1 + m²). Alternatively, since we have the coordinates, we can compute it directly.But let's see. Let me compute |AB|.Given points A(x₁, y₁) and B(x₂, y₂), the distance is:|AB| = √[(x₁ - x₂)² + (y₁ - y₂)²]But since both points lie on the line l, which is x = ty + 1, so x₁ - x₂ = t(y₁ - y₂). Therefore:|AB| = √[ (t(y₁ - y₂))² + (y₁ - y₂)² ] = √[ (t² + 1)(y₁ - y₂)² ] = √(t² + 1) * |y₁ - y₂|So, |AB| = √(t² + 1) * |y₁ - y₂|We already have expressions for y₁ and y₂. Let's compute |y₁ - y₂|.From earlier, y₁ = [-t + √2√(t² + 1)] / (t² + 2)y₂ = [-t - √2√(t² + 1)] / (t² + 2)So, y₁ - y₂ = [ (-t + √2√(t² + 1)) - (-t - √2√(t² + 1)) ] / (t² + 2)= [ (-t + √2√(t² + 1) + t + √2√(t² + 1) ) ] / (t² + 2)= [ 2√2√(t² + 1) ] / (t² + 2)Therefore, |y₁ - y₂| = 2√2√(t² + 1) / (t² + 2)Thus, |AB| = √(t² + 1) * [2√2√(t² + 1) / (t² + 2)] = 2√2 (t² + 1) / (t² + 2)So, |AB| = 2√2 (t² + 1)/(t² + 2)Alright, that's |AB|. Now, we need to find |FE|.Point F is the right focus, which is (1, 0). The line l intersects a circle with diameter FP at another point E. So, first, I need to find the equation of the circle with diameter FP.Points F(1, 0) and P(1, √2/2). Wait, P is (1, √2/2). So, the diameter is from F(1, 0) to P(1, √2/2). Therefore, the circle has endpoints of its diameter at these two points.The center of the circle is the midpoint of FP. Let's compute that.Midpoint M:x-coordinate: (1 + 1)/2 = 1y-coordinate: (0 + √2/2)/2 = √2/4So, center M is (1, √2/4)The radius r is half the distance between F and P.Distance FP:√[(1 - 1)² + (√2/2 - 0)²] = √[0 + ( (√2)/2 )²] = √[ (2)/4 ] = √(1/2) = √2 / 2Therefore, radius r = (√2 / 2)/2 = √2 / 4So, the equation of the circle is:(x - 1)² + (y - √2/4)² = (√2 / 4)² = (2)/16 = 1/8So, the circle equation is:(x - 1)² + (y - √2/4)² = 1/8Now, the line l intersects this circle at point E (other than F). So, we need to find the coordinates of E.We already have the parametric equation of line l: x = ty + 1. Let's substitute this into the circle equation.Substitute x = ty + 1 into the circle equation:(ty + 1 - 1)² + (y - √2/4)² = 1/8Simplify:(ty)² + (y - √2/4)² = 1/8= t² y² + y² - (√2/2) y + ( (√2)/4 )² = 1/8= (t² + 1) y² - (√2/2) y + (2)/16 = 1/8Simplify constants:(2)/16 = 1/8, so:(t² + 1) y² - (√2/2) y + 1/8 = 1/8Subtract 1/8 from both sides:(t² + 1) y² - (√2/2) y = 0Factor out y:y [ (t² + 1) y - √2/2 ] = 0So, solutions are y = 0 and y = (√2/2)/(t² + 1)Since the line passes through F(1, 0), which is one intersection point, the other intersection point E has y-coordinate y = (√2/2)/(t² + 1)Therefore, the coordinates of E are:x = t y + 1 = t*(√2/2)/(t² + 1) + 1 = (t√2/2)/(t² + 1) + 1So, E is ( (t√2/2)/(t² + 1) + 1, (√2/2)/(t² + 1) )Now, we need to find |FE|.Point F is (1, 0), and E is ( (t√2/2)/(t² + 1) + 1, (√2/2)/(t² + 1) )Compute the distance between F and E.Let me denote E's coordinates as (x_E, y_E):x_E = 1 + (t√2/2)/(t² + 1)y_E = (√2/2)/(t² + 1)So, |FE| = √[ (x_E - 1)^2 + (y_E - 0)^2 ] = √[ ( (t√2/2)/(t² + 1) )^2 + ( (√2/2)/(t² + 1) )^2 ]Simplify:= √[ ( (2 t²)/4 ) / (t² + 1)^2 + (2/4 ) / (t² + 1)^2 ]= √[ ( (t²)/2 + 1/2 ) / (t² + 1)^2 ]= √[ ( (t² + 1)/2 ) / (t² + 1)^2 ]= √[ 1 / (2(t² + 1)) ]= 1 / √(2(t² + 1))So, |FE| = 1 / √(2(t² + 1))Alternatively, we can write it as √(1/(2(t² + 1))) or (1/√2) * 1/√(t² + 1)Now, we have |AB| = 2√2 (t² + 1)/(t² + 2) and |FE| = 1 / √(2(t² + 1))Therefore, the product |AB| · |FE| is:2√2 (t² + 1)/(t² + 2) * 1 / √(2(t² + 1)) = 2√2 / √2 * (t² + 1) / (t² + 2) * 1 / √(t² + 1)Simplify:2√2 / √2 = 2So, |AB| · |FE| = 2 * (t² + 1) / (t² + 2) * 1 / √(t² + 1) = 2 * √(t² + 1) / (t² + 2)Wait, let me check that again.Wait, 2√2 / √2 is indeed 2, because √2 / √2 = 1, so 2√2 / √2 = 2.Then, (t² + 1) / √(t² + 1) = √(t² + 1)So, putting it together:|AB| · |FE| = 2 * √(t² + 1) / (t² + 2)So, the expression simplifies to 2√(t² + 1)/(t² + 2)Now, we need to find the maximum value of this expression with respect to t.Let me denote f(t) = 2√(t² + 1)/(t² + 2)We can write this as f(t) = 2√(t² + 1)/(t² + 2)To find its maximum, let's consider f(t)² to make it easier, since the square will achieve maximum at the same t.So, f(t)² = [2√(t² + 1)/(t² + 2)]² = 4(t² + 1)/(t² + 2)²Let me denote u = t² + 1, which is always ≥1 since t² ≥0.Then, f(t)² = 4u / (u + 1)²So, we need to maximize 4u / (u + 1)² for u ≥1.Let me set g(u) = 4u / (u + 1)²Find the maximum of g(u) for u ≥1.Compute derivative of g(u):g'(u) = [4(u + 1)² - 4u * 2(u + 1)] / (u + 1)^4Simplify numerator:4(u + 1)^2 - 8u(u + 1) = 4(u² + 2u + 1) - 8u² - 8u = 4u² + 8u + 4 - 8u² - 8u = (-4u²) + 0u + 4So, g'(u) = (-4u² + 4) / (u + 1)^4 = -4(u² - 1) / (u + 1)^4Set g'(u) = 0:-4(u² - 1) / (u + 1)^4 = 0This occurs when numerator is zero: u² - 1 = 0 => u = ±1But since u = t² +1 ≥1, u =1 is the critical point.Now, check the behavior of g(u):For u >1, g'(u) is negative because u² -1 >0, so numerator is negative, denominator is positive. Therefore, g(u) is decreasing for u >1.At u=1, g(u) = 4*1 / (1 +1)^2 = 4 /4 =1As u approaches infinity, g(u) approaches 4u / u² = 4/u →0Therefore, the maximum of g(u) is at u=1, which is 1.Therefore, the maximum of f(t)² is 1, so the maximum of f(t) is 1.Hence, the maximum value of |AB| · |FE| is 1.Wait, let me verify this because sometimes when we square functions, we might miss something, but in this case, since f(t) is non-negative, the maximum of f(t) occurs at the same point as the maximum of f(t)².So, yes, the maximum value is 1.Therefore, the answer is 1.**Final Answer**The maximum value of (|AB| cdot |FE|) is (boxed{1}).

💡Okay, so I need to find the area of the axial cross-section of a cylinder that's inscribed in a unit cube. The cylinder's axis lies on the diagonal of the cube, and each base of the cylinder touches three faces of the cube at their centers. Hmm, that sounds a bit complicated, but let's break it down step by step.First, let me visualize the cube. A unit cube has sides of length 1. The diagonal of the cube goes from one corner to the opposite corner. If the cylinder's axis is along this diagonal, then the cylinder is oriented in a way that it's passing through the cube from one corner to the other.Now, each base of the cylinder touches three faces of the cube at their centers. The centers of the faces of a unit cube are at the midpoints of each face. For example, on the front face, the center is at (0.5, 0.5, 0), on the right face at (1, 0.5, 0.5), and so on. So, each base of the cylinder is touching three such centers.I think the key here is to figure out the radius of the cylinder's base and its height. Once I have those, I can find the area of the axial cross-section, which I believe is the area of the rectangle formed by the height of the cylinder and the diameter of its base.Let me start by trying to find the radius. Since each base touches three faces at their centers, these centers are points in 3D space. Let me consider one of the bases. Without loss of generality, let's say the base is near the corner (0,0,0). The three centers it touches would be (0.5, 0, 0), (0, 0.5, 0), and (0, 0, 0.5). These three points form a triangle on the base of the cylinder.Wait, so the base of the cylinder is a circle that passes through these three points. That means the circle is the circumcircle of the triangle formed by these three points. So, if I can find the circumradius of this triangle, that will give me the radius of the cylinder's base.Let me find the coordinates of these three points: (0.5, 0, 0), (0, 0.5, 0), and (0, 0, 0.5). Hmm, but these points are in 3D space, but the base of the cylinder is a circle in a plane. I need to figure out the equation of the plane containing these three points.To find the equation of the plane, I can use the general plane equation: ax + by + cz = d. Plugging in the three points:For (0.5, 0, 0): 0.5a + 0 + 0 = d => 0.5a = dFor (0, 0.5, 0): 0 + 0.5b + 0 = d => 0.5b = dFor (0, 0, 0.5): 0 + 0 + 0.5c = d => 0.5c = dSo, from these, we have a = 2d, b = 2d, c = 2d. Let's set d = 1 for simplicity, then a = 2, b = 2, c = 2. So the plane equation is 2x + 2y + 2z = 1, which simplifies to x + y + z = 0.5.Okay, so the base of the cylinder lies on the plane x + y + z = 0.5. Now, I need to find the circumradius of the triangle formed by the points (0.5, 0, 0), (0, 0.5, 0), and (0, 0, 0.5) on this plane.To find the circumradius, I can use the formula for the circumradius of a triangle in 3D space. The formula is R = (abc)/(4Δ), where a, b, c are the lengths of the sides, and Δ is the area of the triangle.First, let's find the lengths of the sides of the triangle.Between (0.5, 0, 0) and (0, 0.5, 0): distance = sqrt[(0.5)^2 + (-0.5)^2 + 0^2] = sqrt[0.25 + 0.25] = sqrt[0.5] = √(1/2) = (√2)/2.Similarly, between (0, 0.5, 0) and (0, 0, 0.5): distance = sqrt[(0)^2 + (-0.5)^2 + (0.5)^2] = sqrt[0 + 0.25 + 0.25] = sqrt[0.5] = (√2)/2.And between (0, 0, 0.5) and (0.5, 0, 0): distance = sqrt[(-0.5)^2 + 0^2 + (-0.5)^2] = sqrt[0.25 + 0 + 0.25] = sqrt[0.5] = (√2)/2.So, all sides are equal, meaning this is an equilateral triangle with side length (√2)/2.Now, the area Δ of an equilateral triangle is (√3/4) * (side length)^2.So, Δ = (√3/4) * ( (√2)/2 )^2 = (√3/4) * (2/4) = (√3/4) * (1/2) = √3/8.Now, the circumradius R is given by (a*b*c)/(4Δ). Since all sides are equal, a = b = c = (√2)/2.So, R = ( (√2)/2 * (√2)/2 * (√2)/2 ) / (4 * √3/8 )Let's compute numerator and denominator separately.Numerator: (√2/2)^3 = ( (√2)^3 ) / 8 = (2√2)/8 = √2/4.Denominator: 4 * (√3/8) = (4√3)/8 = √3/2.So, R = (√2/4) / (√3/2) = (√2/4) * (2/√3) = (√2/2) / √3 = √2/(2√3) = √6/6.So, the radius r of the cylinder's base is √6/6.Okay, that's the radius. Now, let's find the height of the cylinder. The axis of the cylinder is along the space diagonal of the cube. The space diagonal of a unit cube is sqrt(1^2 + 1^2 + 1^2) = sqrt(3). So, the length of the space diagonal is sqrt(3).But the cylinder is inscribed in the cube, so the height of the cylinder can't be the entire space diagonal. Instead, it must fit within the cube such that the bases are touching the centers of the faces.Wait, earlier, we considered one base near (0,0,0), touching (0.5,0,0), (0,0.5,0), and (0,0,0.5). Similarly, the other base would be near the opposite corner (1,1,1), touching (0.5,1,1), (1,0.5,1), and (1,1,0.5).So, the height of the cylinder is the distance between these two planes: x + y + z = 0.5 and x + y + z = 1.5.Wait, but the space diagonal goes from (0,0,0) to (1,1,1). The distance between the two planes x + y + z = 0.5 and x + y + z = 1.5 can be found by the formula for the distance between two parallel planes: |d2 - d1| / sqrt(a^2 + b^2 + c^2). Here, the planes are x + y + z = 0.5 and x + y + z = 1.5, so a = 1, b = 1, c = 1.So, distance = |1.5 - 0.5| / sqrt(1 + 1 + 1) = 1 / sqrt(3) = sqrt(3)/3.Therefore, the height h of the cylinder is sqrt(3)/3.So, now we have the radius r = sqrt(6)/6 and the height h = sqrt(3)/3.The axial cross-section of the cylinder is a rectangle. The width of this rectangle is twice the radius, so 2r = 2*(sqrt(6)/6) = sqrt(6)/3. The height of the rectangle is h = sqrt(3)/3.Therefore, the area of the axial cross-section is width * height = (sqrt(6)/3) * (sqrt(3)/3) = (sqrt(6)*sqrt(3))/9 = sqrt(18)/9.Simplifying sqrt(18) = 3*sqrt(2), so the area is (3*sqrt(2))/9 = sqrt(2)/3.Wait, let me double-check that. sqrt(6)*sqrt(3) is sqrt(18), which is 3*sqrt(2). Then, 3*sqrt(2)/9 simplifies to sqrt(2)/3. Yes, that seems correct.So, the area of the axial cross-section is sqrt(2)/3.**Final Answer**The area of the axial cross-section is boxed{dfrac{sqrt{2}}{3}}.

💡Okay, so I have this equation: (2x^2 + 2y^2 - 10 = 2x + 4y). I need to find the radius of the shape described by this equation. Hmm, I think this is a circle because I see (x^2) and (y^2) terms, which usually indicate a circle, ellipse, hyperbola, or something similar. But since the coefficients of (x^2) and (y^2) are the same, it's probably a circle.First, I should simplify the equation to get it into the standard form of a circle, which is ((x - h)^2 + (y - k)^2 = r^2), where ((h, k)) is the center and (r) is the radius.Let me start by moving all the terms to one side to make it easier to work with. So, I'll subtract (2x) and (4y) from both sides:(2x^2 + 2y^2 - 10 - 2x - 4y = 0)Hmm, that looks a bit messy. Maybe I can factor out the 2 from the (x^2) and (y^2) terms to simplify it further. Let's try that:(2(x^2 - x) + 2(y^2 - 2y) - 10 = 0)Wait, actually, I think I made a mistake there. Let me check. The original equation is (2x^2 + 2y^2 - 10 = 2x + 4y). If I subtract (2x + 4y) from both sides, it should be:(2x^2 + 2y^2 - 2x - 4y - 10 = 0)Yes, that's better. Now, I can factor out the 2 from the (x^2) and (y^2) terms:(2(x^2 - x) + 2(y^2 - 2y) - 10 = 0)Okay, now I have (2(x^2 - x)) and (2(y^2 - 2y)). To complete the square for each, I need to find the right constants to add inside the parentheses so that they become perfect squares.For the (x) terms: (x^2 - x). To complete the square, I take half of the coefficient of (x), which is (-1), so half of that is (-frac{1}{2}), and then square it, which gives (frac{1}{4}). So I need to add (frac{1}{4}) inside the parentheses to complete the square. But since there's a factor of 2 outside, adding (frac{1}{4}) inside is like adding (2 times frac{1}{4} = frac{1}{2}) to the entire equation. I need to balance that by subtracting (frac{1}{2}) somewhere.Similarly, for the (y) terms: (y^2 - 2y). Half of (-2) is (-1), and squaring that gives (1). So I need to add (1) inside the parentheses. Again, since there's a factor of 2 outside, adding (1) inside is like adding (2 times 1 = 2) to the equation. I need to subtract (2) to balance it.Let me write that out step by step:Starting with:(2(x^2 - x) + 2(y^2 - 2y) - 10 = 0)Complete the square for (x):(2left[(x^2 - x + frac{1}{4}) - frac{1}{4}right] + 2(y^2 - 2y) - 10 = 0)Which simplifies to:(2left[(x - frac{1}{2})^2 - frac{1}{4}right] + 2(y^2 - 2y) - 10 = 0)Distribute the 2:(2(x - frac{1}{2})^2 - frac{1}{2} + 2(y^2 - 2y) - 10 = 0)Now, complete the square for (y):(2(x - frac{1}{2})^2 - frac{1}{2} + 2left[(y^2 - 2y + 1) - 1right] - 10 = 0)Simplify:(2(x - frac{1}{2})^2 - frac{1}{2} + 2(y - 1)^2 - 2 - 10 = 0)Combine the constants:(-frac{1}{2} - 2 - 10 = -frac{1}{2} - 12 = -frac{25}{2})So now the equation is:(2(x - frac{1}{2})^2 + 2(y - 1)^2 - frac{25}{2} = 0)Move the constant term to the other side:(2(x - frac{1}{2})^2 + 2(y - 1)^2 = frac{25}{2})To get it into standard form, I can divide both sides by 2:((x - frac{1}{2})^2 + (y - 1)^2 = frac{25}{4})Now, this looks like the standard equation of a circle with center at ((frac{1}{2}, 1)) and radius squared equal to (frac{25}{4}). So the radius (r) is the square root of (frac{25}{4}), which is (frac{5}{2}).Wait, let me double-check that. If I divide both sides by 2, I get:((x - frac{1}{2})^2 + (y - 1)^2 = frac{25}{4})Yes, that's correct. So the radius is (sqrt{frac{25}{4}} = frac{5}{2}).I think that makes sense. Let me recap the steps to make sure I didn't make any mistakes:1. Started with the original equation.2. Moved all terms to one side.3. Factored out the 2 from the (x^2) and (y^2) terms.4. Completed the square for both (x) and (y), adjusting for the constants added inside the parentheses by subtracting the appropriate amounts.5. Simplified and combined constants.6. Divided by 2 to get the equation into standard circle form.7. Identified the radius as the square root of the right-hand side.Everything seems to check out. So the radius of the circle described by the equation is (frac{5}{2}).