14. Let It be the on two circles with diameters (AH) and (BH) lies on the...
heater of ABC Hint B C Signs of i les 14. Let it be the orthocenter of a nondegenerate AABC. Prove that the second point of intersection of two circles with diameters (AHand (BH) lies on the line (AB). distinct points: A and B. Assume (XY) and [X'Y') Ir lies between Y
Let H be the orthocenter of a nondegenerate... 14. Let H be the orthocenter of a nondegenerate & ABC. Prove that the second point of intersection of two circles with diameters (AH] and [BH] lies on the line (AB).
MATH 241 S19 HOMEWORK 14 1. For each of theg, o dx (a) 4x3 + 7xy2 2Уз (b) x,5+1 xy + 1 2. For each of the following, find y': (a). xy sin(xy)-b) cosxy2)-y2+x 3. Find an equation of the line that is tangent to the graph of x2y2 4xy 12y at the point (2,) 4. Show that the graph of xy1 and the graph of x2-y2-1 intersect at right angles. 5. Given that x2 -y2 -1, show that y"...
39. Let L be the minimum length of a ladder that can reach over a fence of height h to a wall located a distance b behind the wall. a. Use Lagrange multipliers to show that L = (h2/3 + 62/3;3/2 (Eigure 20). Hint: Show that the problem amounts to minimizing f (x, y) = (x + b)² + (y+h)? subject to y/b = h/x or ry = bh. b. Show that the value of L is also equal to...
need help Find the length of the curve defined by the parametric equations y3In(t/4)2-1) from t 5 tot- 7 Find the length of parametized curve given by a(t) -0t3 -3t2 + 6t, y(t)1t3 +3t2+ 0t, where t goes from zero to one. Hint: The speed is a quadratic polynomial with integer coefficients. A curve with polar equation 14 7sin θ + 50 cos θ represents a line. Write this line in the given Cartesian form Note: Your answer should be...
11. Find the volume of the given right tetrahedron. (Hint: Consider slices perpendicular to one of the labeled edges.) 3. The solid lies between planes perpendicular to the x-axis at x= -1 and x = 1. The cross-sections perpendicular to the I-axis between these planes are squares whose bases run from the semicircle y = -VI-to the semicircle y = VI- 4. The solid lies between planes perpendicular to the x-axis at x= -1 and .x = 1. The cross-sections...
Fourth Homework (1) Let P-(**.0) and Q ( . (a) Find the pole of the line PQ (b) Find the parametrization of the line PQ (c) Does (ch,顽週lie on the line PQ? 克,2 7, ) lie on the line PQ? (2) Find the distance between the lines (1,0,-1) + t(2,3,0) and m (2,-1,3) +s(0, 1,2). (3) Let A and B be two distinct points of S2. Show that X e I d(X, A) = d(X, b)) is a line and...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2-2.2-b with b є R. (a) Prove that the tangent line of each curve in H at a point (x, y) with y 0 has slope - (b) Let y-f(x) be a...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2b with b ER. (a) Prove that the tangent line of each curve in H at a point (r, y) with y / 0 has slope (b) Let y -f(x) be a...
8) (Problem 17 (a) on page 49) Let p and q be two distinct primes. Show that for any integer a, pq|(a p+q − a p+1 − a q+1 + a 2 ). Hint: Find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo p, and then find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo q. After that, use the following result: Suppose x,...