Find the standard form of the equation of the parabola with the given characteristics. vertex: (5, 25) points on the parabola: (0, 0), (10, 0) D= - X + 10x XFind the standard form of the equation of the hyperbola with the given characteristics. vertices: (+1, 0) asymptotes: y = 17xFind the standard form of the equation of the hyperbola with the given characteristics. ver ices: (8, :4) Foci: (8, i5) A solar collector for heating water is constructed with a sheet of stainless steel that is formed into the shape of a parabola with width w = 18 meters (see figure). The water will flow through a pipe that is located at the focus of the parabola. At what distance from the vertex is the pipe? Consider a particle traveling clockwise on the elliptical path 2 2 X + V = 1. 400 100 The particle leaves the orbit at the point (716, 6) and travels in a straight line tangent to the ellipse. At what point will the particle cross the yaxis? w ) LORAN (long distance radio navigation) for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (185,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, 300 miles apart, are positioned on a rectangular coordinate system at (7150, 0) and (150, 0) and that a ship is traveling on a path with coordinates (x, 75) (see figure). Find the x-coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 500 microseconds (0.0005 second). (Round your answer to one decimal place.) Use a graphing utility to graph the curve represented by the parametric equations. (Indicate the orientation of the curve.) x = 6 sinz'), y = 7 cos(2l9) Eliminate the parameter and write the corresponding rectangular equation. y: 7005 26 X Determine any differences between the curves of the parametric equations. (a) x = t (b) X = c0509) y=6t+4 y=6cos(i9)+4 (C) x = e't (d) x = ef y=5e't+4 y=6ef+4 Are all graphs the same? By eliminating the parameters in (a)7(d), you get y = . Therefore, the graphs 3/ all X the same. Are the orientations and restricted domains the same? 0 "he orientations and restricted domains are the same. 0 "he orientations are the same, but some of the restricted domains are different. 0 "he restricted domains are the same, but some of the orientations are different. Some of the orientations and restricted domains are different. Which of the curves are smooth? (Select all that apply.) C] (a) [1 (b) C] (C) C] (d) X Eliminate the parameter and obtain the standard form of the rectangular equation. Line through (x1, y1) and (x2, 5/2): x = X1 + t(x2 - x1), y = 1/1 + til/2 - V1) Use your result to find a set of parametric equations for the line or conic. (Enter your answers as a commaseparated list.) Line: passes through (0, 0) and (4, *2) Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: x = h + rcosUS'), y = k + r sin(6') Use your result to find a set of parametric equations for the line or conic. (When 0 S 6 5 2n. Enter your answers as a comma? separated list of equations.) Circle: center: (3, 3); radius: 2