Answer to Question #259265 in Analytic Geometry for Tani

Question

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

Answer 1

$$x-2y+z=1=>z=-x+2y+1$$

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

$$\dfrac{x^2+4x+4}{2}-\dfrac{y^2+12y+36}{3} =2+2-12$$

$$\dfrac{(y+6)^2}{24}-\dfrac{(x+2)^2}{16} =1$$

The equation represents the hyperbola.