Answer to Question #155298 in Analytic Geometry for Sourav Mondal

Question

Find the equation of the right circular cone with 

vertex at (1, 2, —1), axis as the x-axis and semi-

vertical angle π/3.

Answer 1

Find the equation of the right circular cone with 

vertex at (1, 2, —1), axis as the x-axis and semi-

vertical angle π/3.

solution:

equation of rigth circular cone is $cos \Theta={ l(x-\alpha)+ m(y-\beta)+ n(z-\Upsilon) \over\sqrt{l^2+m^2+n^2}\sqrt{x^2+y^2+z^2}}$

the directional ratios (l,m,n)=(a,0,0) since the axis is the x-axis

$\theta = {\pi \over 3}={180 \over3} = 60^o$

$\because cos 60^o={a(x-1)+0(y-\beta)+0(z-\gamma) \over \sqrt{x^2 +y^2 + z^2} \sqrt{a^2 + 0^2+0^2}}$

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

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

Square both sides

${1 \over 4} = {(x-1)^2 \over x^2 +y^2 + z^2}$

$x^2 +y^2 + z^2 = 4(x^2 -2x +1)$

$x^2 +y^2 + z^2 = 4x^2 -8x +4$

$3x^2 -y^2-z^2-8x +4=0$

the equation of the right circular cone is

$3x^2 -y^2-z^2-8x +4=0$