A determinant and a surface - 2

Algebra Level pending

Classify the surface r = ( x , y , z ) \mathbf{r} = (x, y, z) , where x x , y y and z z satisfy the determinant equation:

∣ x 1 − y − 2 x z z y 3 ∣ = 7 \begin{vmatrix} x && 1 && -y \\ -2 && x && z \\ z && y && 3 \end{vmatrix} = 7

Hyperboloid of one sheet Plane Ellipsoid The Empty Set Sphere Paraboloid Hyperboloid of two sheets

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

David Vreken
Nov 19, 2020

The determinant calculates to 3 x 2 + z 2 + 2 y 2 + 6 = 7 3x^2 + z^2 + 2y^2 + 6 = 7 , which after rearranging is 3 x 2 + 2 y 2 + z 2 = 1 3x^2 + 2y^2 + z^2 = 1 , an equation of an ellipsoid .

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