### Central Conicoid

• A conicoid is said to be symmetric with the point P if the conicoid is symmetric to the origin when the equation of the conicoid is transformed with origin as the center.

• The origin is the center of the conicoid if ax2 + by2 + cz2 + 2fyz + 2gzx + 2gxy + 2ux + 2vy + 2wz + d = 0 has u = v = w = 0.

• A conicois S given by the equation F(x, y, z) = ax2 + by2 + cz2 + 2fyz + 2gzx + 2gxy + 2ux + 2vy + 2wz + d = 0 has a point P(x0, y0, z0) as the center iff
ax0 + hy0 + gz0 + u = 0
hx0 + by0 + fz0 + v = 0
gx0 + fy0 + cz0 + w = 0

• A conicoid has a central conicoid if it has a unique center, it is non central conicoid if it does not has a center or has more than one center.

• Standard forms of central conicoids
TypeStandard form
coneax2 + by2 + cz2 = 0
imaginary ellipsoidx2/a2 + y2/b2 + z2/c2 = -1
ellipsoidx2/a2 + y2/b2 + z2/c2 = 1
hyperboloid of one sheetx2/a2 + y2/b2 - z2/c2 = 1
x2/a2 - y2/b2 + z2/c2 = 1
-x2/a2 + y2/b2 + z2/c2 = 1
hyperboloid of two sheetsx2/a2 - y2/b2 - z2/c2 = 1
-x2/a2 - y2/b2 + z2/c2 = 1
-x2/a2 + y2/b2 - z2/c2 = 1
• The condition for a line to be a tangent to the central conicoid ax2 + by2 + cz2 = 1 at (x0, y0, z0) is ax0α + by0β + cz0γ = 0, where α, β and γ are the direction ratios of the line.

• The equation of the tangent plane to a central conicoid ax2 + by2 + cz2 = 1 at a point (x0, y0, z0) is axx0 + byy0 + czz0 = 1.

• The condition that the plane ux + vy + wz = p is a tangent to the central conicoid ax2 + by2 + cz2 = 1 is u2/a + v2/b + w2/z = p2

• A planar section of a central conicoid is a conic section.