Q.105. Equation of a Cone.

Question 105.
Find the equation of the cone with vertex (1, 2, 3) having guiding curve which is a circle given by
x^2 + y^2 + z^2 = 4 and x + y + z = 7.

Answer 105.
A cone is a surface generated by the locus of a  straight line which passes through a fixed point and satisfies one more condition, i.e., it  intersects all the points on a given curve called the guiding curve. The fixed point is called the vertex and the straight line in any position is called a generator.

In the above example, the guiding curve is a circle formed by the intersection of
the sphere x^2 + y^2 + z^2 = 4 and the plane x + y + z = 7.

Any line through (1, 2, 3) has equation of the form 
(x - 1) / l = (y - 2) / m = (z - 3) / n ... ( 1 )
Any point on it is P (1 + lr, 2 + mr, 3 + nr)
If the line ( 1) intersects the given curve, the coordinates of P should satisfy the equations of the guiding curve.
=> (1 + lr)^2 + ( 2 + mr)^2 + (3 + nr)^2 = 4   ...   ( 2 ) 
and (1 + lr) + (2 + mr) + (3 + nr) = 7 => (l + m + n) r = 1
Plugging r = 1/(l + m + n) into equation ( 2 ),
[1 + l/(l+m+n)]^2 + [2 + m/(l+m+n)]^2 + [3 + n/(l+m+n)]^2 = 4 ... ( 3 )

From equation ( 1 ),
(x - 1) / l = (y - 2) / m = (z - 3) / n = (x + y + z - 6) / (l + m + n)
=>
l / (l + m + n) = (x - 1) / (x + y + z - 6),
m / (l + m + n) = (y - 2) / (x + y + z - 6) and
n / (l + m + n) = (z - 3) / (x + y + z - 6)

Plugging these in equation ( 3 ),
[1 + (x -1)/(x+y+z-6)]^2 + [2 + (y - 2)/(x+y+z-6)]^2 + [3 + (z - 3)/(x+y+z-6)]^2 = 4
=> (2x+y+z-7)^2 + (2x+3y+2z-14)^2 + (3x+3y+4z-21)^2
= 4(x+y+z-6)^2
is the locus of the line through the vertex which is the required equation of the cone.

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