Analytic Geometry Answers

Find an equation in standard form for the ellipse with the vertical major axis of length 18, and minor axis of length 16.

Find the center, vertices, and foci of the ellipse with equation 3x2 + 6y2 = 18

Find the standard form of the equation of the parabola with a focus at (3, 0) and a directrix at x = -3.

Find the standard form of the equation of the parabola with a focus at (0, 2) and a directrix at y = -2.

Trace the conicoid given by x^2 + 2z^2 = 4y. What are the sections of this
conicoid by the planes x+2=0 and y=1 ?
Describe geometrically.

Find the new equation of the conicoid 9x^2 + 16y^2 - 36z^2 - 36x - 72z = 144 when the coordinate system is changed into a new system with the same origin at (-2, 0, 1) and direction ratios same as the old system.

At what point the origin must be shifted so that linear terms in the conicoid x^2 + 2y^2 - z^2 - 2yz + 2xz + x - 3y + z + 4 = 0 vanish? Justify.

Using projection show that the line passing through (-1, 8, 8) and (6, 2, 0) is perpendicular to the line passing through (4, 2, 3) and (2, 1, 2).

Show that the angle between the two lines in which the plane x - y + 2z = 0 intersects the cone x^2 + y^2 - 4z^2 + 6yz = 0 is tan inverse of (under-root 6 / 7) .

Find the centre and radius of the circle x^2 + y^2 + z^2 + 2x + 2y + 4z = 3, 2x - y - z = 3.