Show that the projection into the xy-plane of the curve of intersection of the parabolic cylinder z=1−3y^2 and the paraboloid z=x^2+y^2 is an ellipse. (a) Find a vector-parametric equation for the ellipse in the xyxy-plane. Shadow: r1(t)=? (b) Find a vector-parametric equation for the curve of intersection of the parabolic cylinder and the paraboloid. Intersection: r2(t)=?

Show that the projection into the xy-plane of the curve of intersection of the parabolic cylinder z=1−3y^2 and the paraboloid z=x^2+y^2 is an ellipse. (a) Find a vector-parametric equation for the ellipse in the xyxy-plane. Shadow: r1(t)=? (b) Find a vector-parametric equation for the curve of intersection of the parabolic cylinder and the paraboloid. Intersection: r2(t)=?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.4: Plane Curves And Parametric Equations

Problem 53E

See similar textbooks

Similar questions

Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?
Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle rolls along the x -axis given that P is at a maximum when x=0.
Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellipsoid 3x2 + 2y2 + 6z2 = 29 at the point (−1, 1, 2).
Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellipsoid 6x2 + 4y2 + 5z2 = 30 at the point (−1, 1, 2). (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)
Find a parametric curve r(t) that represents the curve of intersections of the semi ellipsoid x^2+y^2+4z^2=4 and the cylinder x^2+z^2=1
find a parametrization for the curve. the lower half of the parabola x - 1 = y2
Find a vector parametric equation for the ellipse that lies on the plane 4x - 5y + z = -7 and inside the cylinder (x^2) + (y^2) = 36 for 0 <= u <= 6 and 0 <= v <= 2(pi) where <= is less then equal too where pi = 3.1415
Determine the parametric equations of the curve of intersection of the ellipsoid 4x2 + y2 + 4z2 = 4, and the half-cone y2 = x2 + 2z2, y > 0
Try to sketch by hand the curve of intersection of the parabolic cylinder y = x2 and the top half of the ellipsoid x2 + 2y2 + 2z2 = 4. Then find parametric equations for this curve. (x(t), y(t), z(t)) = for −1.1 ≤ t ≤ 1.1
25) Show that the curve with parametric equations x = tcos(t) , y = tsin(t), lz = t ies on the cone z^2 = x^2 + y^2, and use this fact to help sketch the curve.
At what point on the ellipsoid 2x^2 + y^2 + z^2 = 1 is thetangent plane parallel to the plane x + y + z = 1?
Find all points on the ellipsoid 2x² + 3y² + 4z² = 9at which the plane tangent to the ellipsoid is parallel to the plane x−2y + 3z = 5.