answer:Since F_1 and F_2 are the foci of the ellipse C: frac{x^2}{36} + frac{y^2}{27} = 1, we have F_1(-3, 0) and F_2(3, 0). The semi-major axis a = 6. Point P is on ellipse C with |PF_1| = 8, hence |PF_2| = 4 (according to the property of ellipse). M is the midpoint of line segment PF_1, so the length of line segment OM is: frac{1}{2}|PF_2| = 2. Therefore, the final answer is: boxed{2}. This problem can be solved by determining the coordinates of the ellipse's foci and applying the definition of an ellipse. It primarily tests your understanding of basic properties of ellipses.

answer:The midpoint of begin{pmatrix} 2 9 end{pmatrix} and begin{pmatrix} 8 3 end{pmatrix} is calculated as follows: [ left( frac{2 + 8}{2}, frac{9 + 3}{2} right) = (5, 6). ] This midpoint tells us the normal to the line of reflection is a vector pointing from begin{pmatrix} 2 9 end{pmatrix} to begin{pmatrix} 8 3 end{pmatrix}, which is: [ begin{pmatrix} 8 - 2 3 - 9 end{pmatrix} = begin{pmatrix} 6 -6 end{pmatrix}. ] Normalizing, we simplify to: [ begin{pmatrix} 1 -1 end{pmatrix}. ] Next, calculate the projection of begin{pmatrix} 1 4 end{pmatrix} onto begin{pmatrix} 1 -1 end{pmatrix}: [ operatorname{proj}_{begin{pmatrix} 1 -1 end{pmatrix}} begin{pmatrix} 1 4 end{pmatrix} = frac{begin{pmatrix} 1 4 end{pmatrix} cdot begin{pmatrix} 1 -1 end{pmatrix}}{begin{pmatrix} 1 -1 end{pmatrix} cdot begin{pmatrix} 1 -1 end{pmatrix}} begin{pmatrix} 1 -1 end{pmatrix} = frac{-3}{2} begin{pmatrix} 1 -1 end{pmatrix} = begin{pmatrix} -3/2 3/2 end{pmatrix}. ] Therefore, the reflection of begin{pmatrix} 1 4 end{pmatrix} across begin{pmatrix} 1 -1 end{pmatrix} results in doubling the vector away from the original: [ 2 begin{pmatrix} -3/2 3/2 end{pmatrix} - begin{pmatrix} 1 4 end{pmatrix} = begin{pmatrix} -3 3 end{pmatrix} - begin{pmatrix} 1 4 end{pmatrix} = boxed{begin{pmatrix} -4 -1 end{pmatrix}}. ]

answer:To calculate the hairstylist's weekly earnings, we first need to calculate the daily earnings for each type of haircut and then sum them up. For normal haircuts: 5 normal haircuts/day x 5/normal haircut = 25/day For special haircuts: 3 special haircuts/day x 6/special haircut = 18/day For trendy haircuts: 2 trendy haircuts/day x 8/trendy haircut = 16/day Now, we add the daily earnings from each type of haircut to get the total daily earnings: 25/day (normal) + 18/day (special) + 16/day (trendy) = 59/day Assuming the hairstylist works 7 days a week, we can calculate the weekly earnings: 59/day x 7 days/week = 413/week Therefore, the hairstylist earns boxed{413} per week.

answer:- The slant height of the cone is equal to the radius of the sector, which is 12. - The arc length of the sector is calculated by frac{270^circ}{360^circ} times (2pi times 12) = frac{3}{4} times 24pi = 18pi. - The circumference of the base of the cone equals the arc length, which is 18pi. The radius ( r ) of the base can be calculated using the circumference formula ( 2pi r ), thus ( r = frac{18pi}{2pi} = 9 ). Hence, the cone that can be formed has a base radius of 9 and a slant height of 12, which corresponds to boxed{C}.