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 hence $\frac{x^2}{2} −\frac{y^2}{3}+2x-4y=2.$ It is equivalent to $3x^2-2y^2+12x-24y=12,$ and hence $3(x^2+4x)-2(y^2+12y)=12.$ We conclude that $3(x+2)^2-2(y+6)^2=-48,$ and hence 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.