Check whether or not the conicoid represented by #5x^2+4y^2-4yz+2xz+2x-4y-8z+2=0# is central or not. If it is, transform the equation by shifting the origin to the center. Else, change any one coefficient to make the equation that of a central conicoid.?

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Answer 1 · 2018-02-12T18:09:19

See below.

I hope it helps.

#f(x,y, z) =5x^2+4y^2-4yz+2xz+2x-4y-8z+2=0#

can be represented as

#f(x,y,z) = (p-p_0) cdot M cdot (p-p_0) + C cdot (p-p_0)=0#

with

#p = (x,y,z)#
#M = ((5,0,1),(0,4,-2),(1,-2,0))#
#p_0 = (x_0,y_0,z_0)#

After solving by making #C = (2 sqrt(70/3), 0,-2 sqrt(70/3))# we get

#p_0 = (2/3, 1/6(sqrt(210)-60),1/3(sqrt(210)-13))#

which represents the conicoid center.

From the matrix #M# we can obtain the type of conicoid after solving

#M p = lambda p# or equivalently

#det(M-lambda I_3) = 0# obtaining the eigenvalues

#lambda = (5.39543, 4.57653, -0.971961)# so the conicoid can be reduced after a convenient coordinate change to

#f(X,Y,Z)=5.39543X^2+ 4.57653Y^2-0.971961Z^2+ alpha X + beta Y + gamma Z + delta = 0#

which is a hyperboloid of two sheets added to a plane.