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- (a) Where does the normal line to the ellipse x2 -xy + y2 = 3at the point (-1, 1)intersect theellipse a second time?; (b) Illustrate part (a) by graphing the ellipse and the normalline.Approximate the length of the graph of the ellipse 4x2 + 9y2 = 36 in the first quadrant using Gaussian Quadrature with n = 5. Determine the error in the approximation given that the actual length is 3.7437137.Consider the elipse with equation x^2+4y^2 = 4 a) Find a paramentrization of the ellipse that traverses in a clockwise direction exatly once, starting from (0,1).
- Two friends visit a whispering gallery (in the shape of a semi-ellipsoid) wherethey stand 100 m apart to be at the foci. If one of them is 6 m from thenearest wall, how high is the gallery at its center?4. Find the area of the surface of the revolution formed by rotating the part of the 4x^2 + y^2= 4 ellipse in the first region about the o-y axis.If P1 and P2 are two points on the conic whose angles θ1 and θ2 differ by 180° , show that the harmonic mean of their respectivve values of r is a constant ( you will need to look up what harmonic mean is).
- Consider the hyperboloid of two sheets of equation x23−y23−z22=1. Mark the alternative that contains the value of k>0 for which the major axis of the ellipse obtained by the intersection of the quadric with the x=k plane has length 3.Find a parametrization for the following curve. a.) The ellipse centered at (2,-3) whose major axis is horizontal and of length 4 while the minor axis is of length 3.The equation x2 - xy + y2 =3 represents a “rotatedellipse,” that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipsecrosses the -axis and show that the tangent lines at thesepoints are parallel.