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- Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many times, but the underlying space curve intersects the xy-plane only twice.A particle moves in the xy-plane in such a way that its path is defined by x = et cos t and y = et sin 2t. Find the speed of the particle when t = /2.Find a parametric description for the circle x2+y2-6y=0, oriented clockwise and where t begins at 0. Restrict the domain of t such that one full revolution occurs.
- Write the parametrization for a circle with each of the following conditions: Centered at the origin; initial point (t=0) is (-7,0)and one full circle is traced out counterclockwise from t=0 to t=piShow that the tangent line at a point P = (x0, y0) on the hyperbola(x/a)2−(y/b)2= 1 has equation Ax − By = 1 where A = x0 a2 and B = y0b2 .Consider the right triangle withvertices (0, 0), (0, b), and (a, 0), where a > 0 and b > 0. Showthat the average vertical distance from points on the x-axis to thehypotenuse is b/2, for all a > 0.