Answer to Question #133690 in Analytic Geometry for Sourav mondal

Question #133690
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)
1
Expert's answer
2020-09-21T14:37:44-0400

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

xxayyazzaxbxaybyazbzaxcxaycyazcza\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= 0


x3y0z0002000000010\begin{vmatrix} x-3 & y-0 & z-0\\ 0-0 & 2-0 & 0-0\\ 0-0 & 0-0 & 1-0\\ \end{vmatrix} =0=0


Plane S equation:

2x+3y+6z6=02*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)

x02=y03=z06\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=3tx=3*t

Find the angle between the straight lines A and D

ϕ=73.381°\phi = 73.381°

Length of line OH is 0.85:

tg(ϕ)=HA0.85tg(\phi)=\dfrac{HA}{0.85}

HA=2.78HA = 2.78

Equation of cone

x22.782+y22.782z20.852=0\dfrac{x^2}{2.78^2}+\dfrac{y^2}{2.78^2}-\dfrac{z^2}{0.85^2}=0


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