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MTH 166, Homework #10, Fall 2017Name ________________________________________Instructions: Write your work up neatly and attach to this page. Record your final answers (only) directly on this page if they are short; if too long indicate which page of the work the answer is on and mark it clearly. Use exact values unless specifically asked to round.Plot each number in the complex plane and find its magnitude.z=4ib. 2+3ic. -3-4id. 3-ie. 2+5if. 2Write each complex number in polar form.2+2ib. -2+2i3c. -2+3id. 1-i5Write each complex number in rectangular form.6cosπ6+isinπ6c. 8cos7π4+isin7π45cosπ2+isinπ2d. 20cos205°+isin205°Find z1z2 and z1z2 of the complex numbers.z1=6cos20°+isin20°, z2=5cos50°+isin50°z1=3cos5π8+isin5π8, z2=4cosπ16+isinπ16z1=1+i, z2=-1+iz1=1+i, z2=2-3iUse DeMoivre’s Theorem to find the indicated power.2cos15°+isin15°3d. 3cosπ4+isinπ4512 cos5π18+isin5π186e. 2-i71+i4Find all the complex plex square roots of 9cos5π3+isin5π3Complex cube roots of 27cos306°+isin306°Complex fourth roots of 4cos4π3+isin4π3Complex fifth roots of 1+iComplex cube, fourth, fifth and sixth roots of 1Complex cube, fourth, fifth and sixth roots of -iGraph the conic sections. Find and label any vertices, center, foci, asymptotes, directrix, etc.x216+y24=1m. x216+y249=1x28-y225=1n. y225-x264=1y2=-8xo. 4x2+25y2=100x2=1-4y2p. 9y2-25x2=225y=±x2-3q. 8y2+4x=0x-229+y-124=1r. x-122+y+325=1x+3225-y216=1s. x-22=8y-1y+42=12x+2t. x-32+4y-22=16x-32-4y+32=4u. 9x2+16y2-18x+64y-71=04x2+y2+16x-6y-39=0v. 16x2-y2+64x-2y+67=09x2-16y2-36x-64y+116=0w. x2+6x+8y+1=0y2-2y+12x-35=0Graph the conic section with the given information and write the equation of the graph in standard form.Ellipse: foci -5,0,5,0, vertices -8,0,8,0Ellipse: foci 0,-2,(0,2), x-intercepts -2,2Ellipse: major axis length 10, minor axis length 4, center (-2,3)Ellipse: vertices 7,9,(7,3), minor axis endpoints 5,6,(9,6)Hyperbola: foci 0,-3,(0,3), vertices 0,-1,(0,1)Hyperbola: endpoints of transverse axis 0,-6,0,6, asymptotes y=±2xHyperbola: center (4,-2), focus (7,-2), vertex (6,-2)Parabola: focus (7,0), directrix x=-7Parabola: focus 0,15, directrix y=-15Parabola: vertex 5,-2, focus (7,-2)Parabola: focus (2,4), directrix x=-4Parabola: focus (7,7), directrix y=-9For each of the graphs of conic sections below, write the equation of the graph in standard form.The elliptical ceiling in Statuary Hall in the U.S. Capitol Building is 96 ft long and 23 feet wide. If the origin is the center of the semi-elliptical room, write an equation for the shape of the dome.John Quincy Adams discovered that he could overhear the conversations of opposing party leaders near the left side of the chamber if he situated his desk at the focus on the right side of the chamber. How far from the center, along the major axis, did Adams situate his desk (to the nearest foot).An Earth satellite has an elliptical orbit described by x250002+y247502=1 with units in miles. The center of the Earth is at (16,0). The perigee of the satellite’s orbit is the point nearest to Earth’s center. If the radius of the Earth is approximately 4000 miles, find the distance to the perigee above the Earth’s surface.The apogee of the satellite’s orbit is the points furthest from the Earth’s center. Find the distance of the apogee above the surface.Scattering experiments are experiments in which moving particles are deflected by various forces, led to the concept of the nucleus of an atom. In 1911, Ernest Rutherford discovered that when alpha particles are directed along hyperbolic paths. If a particle gets as close as 3 units to the nucleus along a hyperbolic path with asymptotes given by y=12x, what is the equation of its path?The towers of a suspension bridge are 800 ft apart and rise 160 ft above the road. The cable between the towers has (approx.) the shape of a parabola, and the cable just touches the road midway between the towers. What is the height of the cable 100 ft from a tower?Graph the parametric plane curve on the indicated interval. Use arrows to show the orientation of the curve. Write the equation in non-parametric form.x=t+2, y=t2, -2≤t≤2x=t, y=t-1, t≥0x=2t, y=t-1, -∞,∞x=2sint,y=2cost, 0≤t≤2πx=2cost, y=3sint, 0≤t≤2πx=sect, y=tant, -π2<t<π2Write the parametric equations for the given equation, or fitting the given description.y=4x-3y=x2-3Circle with center at (3,5), Radius is 6Ellipse with center at (-2,3), major axis is 10, minor axis is 4Hyperbola with center at origin, and vertices at 0,±4, foci at 0,±5Line passing through -2,4 and 1,7Identify the conic section. Sketch the graph.r=31+sinθb. r=122-4cosθc. r=153-2cosθd. r=124+5sinθ ................
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