answer:1. Define the variables: [ begin{aligned} &text{Let } a = text{number of blue-eyed brunettes}, &text{Let } b = text{number of blue-eyed blondes}, &text{Let } c = text{number of green-eyed blondes}, &text{Let } d = text{number of green-eyed brunettes}. end{aligned} ] 2. Given that 65% of blue-eyed individuals are brunettes: [ frac{a}{a+b} = 0.65 ] Solving for the ratio ( frac{b}{a} ): [ begin{aligned} &frac{a}{a+b} = 0.65 implies frac{a+b}{a} = frac{1}{0.65} = frac{20}{13}, &frac{b}{a} = frac{20}{13} - 1 = frac{7}{13}. end{aligned} ] 3. Given that 70% of blondes have blue eyes: [ frac{b}{b+c} = 0.7 ] Solving for the ratio ( frac{c}{b} ): [ begin{aligned} &frac{b}{b+c} = 0.7 implies frac{b+c}{b} = frac{1}{0.7} = frac{10}{7}, &frac{c}{b} = frac{10}{7} - 1 = frac{3}{7}. end{aligned} ] Then, finding ( frac{c}{a} ): [ frac{c}{a} = frac{c}{b} cdot frac{b}{a} = frac{3}{7} cdot frac{7}{13} = frac{3}{13}. ] 4. Given that 10% of green-eyed individuals are blondes: [ frac{c}{c+d} = 0.1 ] Solving for the ratio ( frac{d}{c} ): [ begin{aligned} &frac{c}{c+d} = 0.1 implies frac{c+d}{c} = frac{1}{0.1} = 10, &frac{d}{c} = 10 - 1 = 9. end{aligned} ] Then, finding ( frac{d}{a} ): [ frac{d}{a} = frac{d}{c} cdot frac{c}{a} = 9 cdot frac{3}{13} = frac{27}{13}. ] 5. Compute the total population ratio ( frac{a+b+c+d}{a} ): [ begin{aligned} &frac{a+b+c+d}{a} = 1 + frac{b}{a} + frac{c}{a} + frac{d}{a}, &frac{a+b+c+d}{a} = 1 + frac{7}{13} + frac{3}{13} + frac{27}{13}, &frac{a+b+c+d}{a} = 1 + frac{37}{13} = frac{13}{13} + frac{37}{13} = frac{50}{13}. end{aligned} ] 6. Calculate the fraction of green-eyed brunettes in the entire population: [ begin{aligned} &frac{d}{a+b+c+d} = frac{d}{a} cdot frac{a}{a+b+c+d}, &frac{d}{a+b+c+d} = frac{27}{13} cdot frac{13}{50} = frac{27}{50} = 0.54 = 54%. end{aligned} ] Conclusion: [ boxed{54%} ]

answer:1. Substitute y = x^2 + 4 into the hyperbola equation: [ (x^2 + 4)^2 - mx^2 = 4. ] Simplifying, we get: [ x^4 + 8x^2 + 16 - mx^2 = 4. ] [ x^4 + (8-m)x^2 + 12 = 0. ] 2. For tangency, the discriminant of this quadratic in x^2 must be zero: [ (8-m)^2 - 4 cdot 1 cdot 12 = 0. ] [ 64 - 16m + m^2 - 48 = 0. ] [ m^2 - 16m + 16 = 0. ] [ (m - 8)^2 = 0. ] [ m = 8. ] 3. Verify m = 8 works by checking y values: [ y = x^2 + 4 geq 4. ] For m = 8, the hyperbola equation becomes y^2 - 8x^2 = 4 or (y^2 - 4) = 8x^2, ensuring non-negative y values. Conclusion: The value of m such that the given parabola is tangent to the hyperbola is boxed{8}.

answer:To solve for the directrix of the parabola given by the equation x^{2}=4y, we start by identifying the standard form of a parabola that opens upwards or downwards, which is x^2 = 4py. Here, comparing this with our given equation, we can see that 4p = 4, which leads us to find the value of p. Starting with the equation of the parabola: [x^2 = 4y] We identify 4p from the equation, which is equal to 4. Thus, we have: [4p = 4] Solving for p, we divide both sides by 4: [p = frac{4}{4}] This simplifies to: [p = 1] However, to find the directrix of a parabola, we use the formula y = -frac{p}{2} when the parabola opens upwards (since the vertex form of the parabola is at the origin and the focus is above the vertex). Given p = 1, we substitute p into the formula for the directrix: [y = -frac{1}{2}] This simplification was incorrect in the initial solution; the correct step should be recognizing that the directrix is located a distance p below the vertex (since the vertex is at the origin, (0,0)). Therefore, the correct calculation for the directrix, given p = 1, is: [y = -p] Substituting p = 1 into the equation gives us: [y = -1] Therefore, the correct equation for the directrix of the parabola x^2 = 4y is y = -1. This corresponds to option D in the multiple-choice question. Thus, the final answer, following the formatting rules, is: [boxed{D}]

answer:1. **Identify the given information and setup the problem:** - In the right triangle ( triangle ABC ), ( angle BAC = 90^circ ) and ( angle ABC = 54^circ ). - Point ( M ) is the midpoint of the hypotenuse ( BC ). - Point ( D ) is the foot of the angle bisector drawn from vertex ( C ). - ( AM cap CD = {E} ). - We need to prove that ( AB = CE ). 2. **Use vector notation to express the points:** - Let ( A = vec{A} ), ( B = vec{B} ), and ( C = vec{C} ). - Since ( M ) is the midpoint of ( BC ), we have: [ vec{M} = frac{vec{B} + vec{C}}{2} ] 3. **Express the coordinates of ( D ) using the angle bisector theorem:** - The angle bisector theorem states that the ratio of the segments created by the angle bisector is equal to the ratio of the other two sides of the triangle. - Since ( angle BAC = 90^circ ), ( angle ACB = 36^circ ) (since ( angle ABC = 54^circ )). - Therefore, the ratio ( frac{BD}{DC} = frac{AB}{AC} ). 4. **Use the coordinates of ( D ) and the properties of the angle bisector:** - Let ( D ) divide ( BC ) in the ratio ( frac{AB}{AC} ). - Using the coordinates, we have: [ vec{D} = frac{AB cdot vec{C} + AC cdot vec{B}}{AB + AC} ] 5. **Find the coordinates of ( E ) using the intersection of ( AM ) and ( CD ):** - The line ( AM ) can be parameterized as: [ vec{R}(t) = vec{A} + t(vec{M} - vec{A}) ] - The line ( CD ) can be parameterized as: [ vec{S}(u) = vec{C} + u(vec{D} - vec{C}) ] - Set ( vec{R}(t) = vec{S}(u) ) to find the intersection point ( E ). 6. **Solve for ( t ) and ( u ) to find the coordinates of ( E ):** - Equate the parameterized equations and solve for ( t ) and ( u ): [ vec{A} + t(vec{M} - vec{A}) = vec{C} + u(vec{D} - vec{C}) ] 7. **Use trigonometric identities to simplify the expressions:** - Use the given angles and trigonometric identities to simplify the expressions for ( CE ) and ( AB ). - Note that ( cos(36^circ) = frac{sqrt{5} + 1}{4} ) and ( sin(36^circ) = frac{sqrt{10 - 2sqrt{5}}}{4} ). 8. **Prove that ( AB = CE ):** - Using the simplified expressions, show that: [ frac{CE}{AB} = 1 ] - This implies ( CE = AB ). The final answer is ( boxed{ AB = CE } ).