# 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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Feb 12, 2018

See below.

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$f \left(x , y , z\right) = 5 {x}^{2} + 4 {y}^{2} - 4 y z + 2 x z + 2 x - 4 y - 8 z + 2 = 0$

can be represented as

$f \left(x , y , z\right) = \left(p - {p}_{0}\right) \cdot M \cdot \left(p - {p}_{0}\right) + C \cdot \left(p - {p}_{0}\right) = 0$

with

$p = \left(x , y , z\right)$
$M = \left(\begin{matrix}5 & 0 & 1 \\ 0 & 4 & - 2 \\ 1 & - 2 & 0\end{matrix}\right)$
${p}_{0} = \left({x}_{0} , {y}_{0} , {z}_{0}\right)$

After solving by making $C = \left(2 \sqrt{\frac{70}{3}} , 0 , - 2 \sqrt{\frac{70}{3}}\right)$ we get

${p}_{0} = \left(\frac{2}{3} , \frac{1}{6} \left(\sqrt{210} - 60\right) , \frac{1}{3} \left(\sqrt{210} - 13\right)\right)$

which represents the conicoid center.

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

$M p = \lambda p$ or equivalently

$\det \left(M - \lambda {I}_{3}\right) = 0$ obtaining the eigenvalues

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

$f \left(X , Y , Z\right) = 5.39543 {X}^{2} + 4.57653 {Y}^{2} - 0.971961 {Z}^{2} + \alpha X + \beta Y + \gamma Z + \delta = 0$

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

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