Question

Find the section of the conicoid (x² /2) − (y ²/ 3 )= 2z by the plane x− 2y+ z = 1. What conic does this section represent? Justify your answer.

Answer 1

Let us find the section of the conicoid $\frac{x^2}{2} −\frac{y^2}{3}= 2z$  by the plane x− 2y+ z = 1.

It follows that $\frac{x^2}{2} −\frac{y^2}{3}= 2(1-x+2y),$ and therefore $\frac{x^2}{2} −\frac{y^2}{3}+2x-4y-2=0.$ The last equality equivalent to $3x^2-2y^2+12x-24y-12=0,$ and thus $3(x^2+4x)-2(y^2+12y)-12=0.$ It follows that $3(x+2)^2-2(y+6)^2=-48,$ and consequently the equation of the section is $\frac{(y+6)^2}{24}-\frac{(x+2)^2}{16}=1.$

It follows that this section represents a hyperbola.