Answer to Question #45072 in Analytic Geometry for jasvinder

Question

Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example.
(i) Any line through the origin cuts the sphere x^2+y^2+z^2=4 at exactly two points.
(ii) The plane making intercept at the z-axis and parallel to the xy-plane intersects the cone x^2+y^2 = z^2(tan theta)^2 in a circle.
(iii) There exists no line with 1/under-root3 ,1/under-root2 ,1/under-root6 as direction cosines.
(iv) The tangent planes at the extremities of any axis of an ellipsoid are perpendicular.
(v) A section of an elliptic paraboloid by a plane is always an ellipse.
(vi) The curve xy^2+yx^2=0 is symmetric about the origin.
(vii) There exists a unique line which is perpendicular to the lines x=y=z/2 and x=y= -z.
(viii) The plane 3x+4y+2z=1 touches the conicoid 3x^2+2y^2=z^2=1.
(ix) The xy- plane intersects the sphere x^2+y^2+z^2+2x-z=2 in a great circle.
(x) Non degenerate conics are non-central.