Question

Find the equation of the right circular cone whose vertex is (2, 1, 0), semivertical angle is 30° and the axis is the line x-2/3=y-1/1=z/2

Answer 1

Direction ratio of generator is ( (x - 2), (y - 1), z).

Direction ratio of axis is (3, 1, 2).

Semi vertical angle is 30°.

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

$\dfrac{\sqrt{3}}{2}=\dfrac{(3x+y+2z-7)}{(\sqrt{14}.\sqrt{(x-2)^{2}+(y-1)^{2}+z^{2}}}$

$42[(x-2)^{2}+(y-1)^2+z^{2}]=4[(3x+y+2z-7)^2]$

$6x^2+38y^2+16z^2-24xy-16yz-48xz-28y+112z+14=0$ is the equation of the cone.