Answer to Question #203631 in Analytic Geometry for Om kaith

Question

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 45◦ .

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 is $2, 1, -2.$

Thus, 

$$\cos 45\degree=\dfrac{2(x-1)+1(y+1)-2(z-2)}{\sqrt{(x-1)^2+(y+1)^2+(z-2)^2}\sqrt{(2)^2+(1)^2+(-2)^2}}$$

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

On simplifying this we get

$$9x^2-18x+9+9y^2+18y+9+9z^2-36z+36$$

$$=8xy-16xz+8x^2+24x+2y^2+12y-8yz$$

$$+8z^2-24z+18$$

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