Question

Find the section of the conicoid x^2/2-2y^2/3 = 2z by the plane x-2y+z = 1. What conic does this section represent? Justify your answer.

Answer 1

$z=2y-x+1$

$x^2/2-2y^2/3=2(2y-x+1)$

$3x^2-4y^2=24y-12x+12$

$3(x^2+4x+4)-12-4(y^2+6y+9)+36=12$

$3(x+2)^2-4(y+3)^2=-12$

$\frac{(x+2)^2}{4}-\frac{(y+3)^2}{3}=-1$

This is a equation of conjugated hyperbola.

centre: $(-2,-3)$

semi-axes:

$a=2,b=\sqrt 3$