Question

find the equation of cone generated by lines drawn through the origin to meet the circle through the points (3,0,0) ,(0,2,0),(0,0,1)

Answer 1

Let find a plane S containing points A(3,0,0) ,B(0,2,0),C(0,0,1):

$\begin{vmatrix} x-x_a & y-y_a & z-z_a\\ x_b-x_a & y_b-y_a & z_b-z_a\\ x_c-x_a & y_c-y_a & z_c-z_a\\ \end{vmatrix}$ $= 0$

$\begin{vmatrix} x-3 & y-0 & z-0\\ 0-0 & 2-0 & 0-0\\ 0-0 & 0-0 & 1-0\\ \end{vmatrix}$ $=0$

Plane S equation:

$2*x+3*y+6*z-6=0$

Let find a height OH of a cone. OH is perpendicular to the plane S and passes through point O(0,0,0)

$\dfrac{x-0}{2}=\dfrac{y-0}{3}=\dfrac{z-0}{6}$

Let us find the equation of the straight line passing through the origin and point A

As a result, the parametric equation of the straight line D was obtained

$x=3*t$

Find the angle between the straight lines A and D

$\phi = 73.381°$

Length of line OH is 0.85:

$tg(\phi)=\dfrac{HA}{0.85}$

$HA = 2.78$

Equation of cone

$\dfrac{x^2}{2.78^2}+\dfrac{y^2}{2.78^2}-\dfrac{z^2}{0.85^2}=0$