An equation of a hyperbola is given.
25x2 − 16y2 = 400
(a) Find the vertices, foci, and asymptotes of the hyperbola.
(Enter your asymptotes as a comma-separated list of equations.)
vertex
(x, y)
=
(smaller x-value)
vertex
(x, y)
=
(larger x-value)
focus
(x, y)
=
(smaller x-value)
focus
(x, y)
=
(larger x-value)
asymptotes
(b) Determine the length of the transverse axis.
(c) Sketch a graph of the hyperbola.
(a) Find the equation of a plane π that contains the line in the
intersection of the planes x+y+3z =
2andx−y+z=1andtheorigin. Howmanysuchplanesarethere?
(b) What if instead of the origin we ask for the plane
containing the same line and the point
(0, −1/4, 3/4)? What changes?
(c) Back to the plane π you found in item a, let Rπ be the
reflection across π. Pick an explicit
An equation of an ellipse is given. 2x^2 + 64y^2 = 128 (a) Find
the vertices, foci, and eccentricity of the ellipse. vertex(x, y)=
(smaller x-value) vertex(x, y)= (larger x-value) focus(x, y)=
(smaller x value) focu (x, y)= (larger x-value) eccentricity (b)
Determine the length of the major axis. Determine the length of the
minor axis. (c) Sketch a graph of the ellipse.
1. Find an equation for the line in the xy−plane that is tangent
to the curve at the point corresponding to the given value of t.
Also, find the value of d^2y/dx^2 at this point. x=sec t, y=tan t,
t=π/6
2. Find the length of the parametric curve: x=cos t, y=t+sin t,
0 ≤ t ≤ π. Hint:To integrate , use the identity, and
complete the integral.
A) Find the equation of the plane that passes through (2, -1,3) and is perpendicular to the line x = 2-3t, y = 3 + t, z = 5t
B) Find the equation where the planes 2x-3y + z = 5 and x + y-z = 2 intersect.
C) Find the distance from the point (2,3,1) to the x + y-z = 2 plane.
D) Find the angle between the planes x + y + z = 1 and x-2y...
Find the equation of the tangent plane and the
parametric equations for the normal line to the surface
x2 + y2 - z = 0 at the point P(4,-1, 6).
Show all steps