Answer to Question #91738 in Analytic Geometry for Ra

Question

To find the equation of the right circular cone whose vertex is (1,-1,2), the axis is
x-1/2 = y+1/1 = z-2/-2
and the semi-vertical angle is 45º

Answer 1

Let P(x,y,z) be any point on the cone. The direction ratio from the origin(axis) to point on the cone is x-1, y+1, z-2 and the direction ratio of the axis are 2,1,-2.

Thus, $Cos45^0=[{(x-1)2+(y+1)1+(z-2)(-2)}]/[\sqrt{\smash[b]{(x-1)^2+(y+1)^2+(z-2)^2}}\sqrt{\smash[b]{4+1+4}}]$

$$1/\sqrt{\smash[b]{2}}=[{(x-1)2+(y+1)1+(z-2)(-2)}]/[\sqrt{\smash[b]{(x-1)^2+(y+1)^2+(z-2)^2}}\sqrt{\smash[b]{4+1+4}}]$$

On simplifying this we get

$$x^2+7y^2+z^2-8xy+8yz+16xz+6y-12z-42x+36=0$$