Answer to Question #91738 in Analytic Geometry for Ra

Question #91738
To find the equation of the right circular cone whose vertex is (1,-1,2), the axis is
x-1/2 = y+1/1 = z-2/-2
and the semi-vertical angle is 45º
1
Expert's answer
2019-07-18T13:00:06-0400

Let P(x,y,z) be any point on the cone. The direction ratio from the origin(axis) to point on the cone is x-1, y+1, z-2 and the direction ratio of the axis are 2,1,-2.

Thus, Cos450=[(x1)2+(y+1)1+(z2)(2)]/[(x1)2+(y+1)2+(z2)24+1+4]Cos45^0=[{(x-1)2+(y+1)1+(z-2)(-2)}]/[\sqrt{\smash[b]{(x-1)^2+(y+1)^2+(z-2)^2}}\sqrt{\smash[b]{4+1+4}}]

1/2=[(x1)2+(y+1)1+(z2)(2)]/[(x1)2+(y+1)2+(z2)24+1+4]1/\sqrt{\smash[b]{2}}=[{(x-1)2+(y+1)1+(z-2)(-2)}]/[\sqrt{\smash[b]{(x-1)^2+(y+1)^2+(z-2)^2}}\sqrt{\smash[b]{4+1+4}}]

On simplifying this we get


x2+7y2+z28xy+8yz+16xz+6y12z42x+36=0x^2+7y^2+z^2-8xy+8yz+16xz+6y-12z-42x+36=0


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