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 are $2,1,-2.$

Given the semi-vertical angle is 45◦

$$\cos 45\degree=\dfrac{1}{\sqrt{2}}$$

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

$$2(2x+y-2z+3)^2=9((x-1)^2+(y+1)^2+(z-2)^2)$$

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

$$+8z^2-24z+18=9x^2-18x+9$$

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

On simplifying this we get

$$x^2+7y^2+z^2-8xy+8yz+16xz$$

$$-42x+6y-12z+36=0$$