Question

Find the equation of the right circular cone whose vertex is (1,0,1), the axis is
x−1 = y−2 = z−3, and the semi-vertical angle is 30^◦. Also, find the section of
the cone by the coordinate planes.

Answer 1

Direction ratio of generator is ((x-1),y,(z-1)).

Direction ratio of axis is (1,1,1).

Half angle is 30°.

$cos (30°)=(a_1a_2+b_1b_2+c_1c_2)/(\sqrt{a_1^{2}+b_1^{2}+c_1^{2}}.\sqrt{a_2^{2}+b_2^{2}+c_2^{2}})$

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

$\implies 9[(x-1)^{2}+y^2+(z-1)^{2}]=4[(x+y+z-2)^2]$

$\implies 5x^2+5y^2+5z^2-8xy-8yz-8xz-2x-2z+16y+2=0$

is the equation of the cone.

Section of the cone by $x=0$ is $5y^2+5z^2-8yz-2z+16y+2=0$.

Section of the cone by $y=0$ is $5x^2+5z^2-8xz-2x-2z+2=0$ .

Section of the cone by $z=0$ is $5x^2+5y^2-8xy-2x+16y+2=0$.