define the enveloping cone
Answers
Enveloping come is a line of any curve which rotate or moves about fixed point gives a shape cone
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Answer:Enveloping cone is nothing but a line of any curve which rotate or moves about fixed point gives a shape cone.
Step-by-step explanation:
Equation of a cone, Enveloping cone of a sphere, Ring circular cone. i.e r2(a(l)2+b(m)2+c(n)2+2hlm+2gln +2fmn)=0 But r2 ≠0, ∴ al2+bm2+cn2+2hlm+2gln +2fmn =0 i.e D.
eg:-
Q:-What is the enveloping cone of shere: _ x^2+ y^2 +z^2-2x+4z=1 with vertex (1,1)?
sol:-
Given,
S= x² + y² + z² - 2x + 4z - 1 →(i)
Comparing with general equation
-S= x² + y² + z² +2ux +2vy +2wz +d
We get,
2u = -2 => u = - 12v =0 => v =02w =4 => w =2
Given vertex is (1,1,1) = (a, b, c)
Now the equation of enveloping cone is given bySS1 =T²HereS1 = (1)² +(1)² +(1)² - 2.(1) + 4(1) - 1= 4 (putting the values of a, b, c in S)
T = tangent plane=xa + yb + zc +u(x+a)+v(y+b) +w(z+c) +d= x + y + z - (x+1) + 2(z+1) - 1= y +3z
Therefore,Equation of the enveloping coneSS1 = T²=>(x² + y² + z² - 2x + 4z - 1)(4) =(y+3z)²=>4x² +4y² +4z² - 8x +16z - 4 = y² +6yz +9z²=>4x² + 3y² - 5z² - 8x + 16z - 6yz - 4 =0