Answer to Question #203631 in Analytic Geometry for Om kaith

Question #203631

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 45◦ .



1
Expert's answer
2021-06-07T18:43:38-0400

Let P(x,y,z)P(x, y, z) be any point on the cone. The direction ratio from the origin(axis) to point on the cone is x1,y+1,z2x-1, y+1, z-2 and the direction ratio of the axis is 2,1,2.2, 1, -2.

Thus, 


cos45°=2(x1)+1(y+1)2(z2)(x1)2+(y+1)2+(z2)2(2)2+(1)2+(2)2\cos 45\degree=\dfrac{2(x-1)+1(y+1)-2(z-2)}{\sqrt{(x-1)^2+(y+1)^2+(z-2)^2}\sqrt{(2)^2+(1)^2+(-2)^2}}


12=2x+y2z+33(x1)2+(y+1)2+(z2)2\dfrac{1}{\sqrt{2}}=\dfrac{2x+y-2z+3}{3\sqrt{(x-1)^2+(y+1)^2+(z-2)^2}}

On simplifying this we get


9x218x+9+9y2+18y+9+9z236z+369x^2-18x+9+9y^2+18y+9+9z^2-36z+36

=8xy16xz+8x2+24x+2y2+12y8yz=8xy-16xz+8x^2+24x+2y^2+12y-8yz

+8z224z+18+8z^2-24z+18

x2+7y2+z28xy+16xz+8yz42x+6y12z+36=0x^2+7y^2+z^2-8xy+16xz+8yz-42x+6y-12z+36=0


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