Question

Find the equation of the cone with vertex at (1,1,1) guiding curve x2 + y2 +z2=1,x+y+z=1.​

Answer 1

The equation of the cone is given by the equation :

$(1 - R^2) \times (x - 1)^2 + (1 - R^2) \times (y - 1)^2 - (1 + R^2) \times (z - 1)^2 = 0$

To find the equation of the cone with a vertex at (1, 1, 1) and the guiding curve given by $x^2 + y^2 + z^2 = 1$  and $x + y + z = 1$, we can use the general equation of a cone and substitute the given information.

The general equation of a cone with vertex (h, k, l) can be expressed as:

$(x - h)^2 + (y - k)^2 + (z - l)^2 = A^2 \times [ (x - h)\div a ]^2 + [ (y - k)\div b ]^2 + [ (z - l)\div c ]^2$

where (a, b, c) represents the direction ratios or guiding vector of the cone, and A is a constant representing the radius of the cone.

Given:

Vertex: (h, k, l) = (1, 1, 1)

Guiding curve 1: $x^2 + y^2 + z^2 = 1$

Guiding curve 2: $x + y + z = 1$

From guiding curve 2, we can express z in terms of x and y:

z = 1 - x - y

Substituting this value of z in guiding curve 1:

$x^2 + y^2 + (1 - x - y)^2 = 1\\x^2 + y^2 + 1 - 2x - 2y + x^2 + y^2 = 1\\2x^2 + 2y^2 - 2x - 2y = 0\\x^2 + y^2 - x - y = 0$

Now, we can compare this equation with the general equation of the cone to find the direction ratios (a, b, c).

Comparing the coefficients, we have:

$a^2 = 1, b^2 = 1, c^2 = -1$

Since $c^2$ is negative, it indicates that the cone is pointing downward.

Taking the square roots, we have:

$a = 1, b = 1, c = -1$

Substituting the values into the general equation of the cone, we get:

$(x - 1)^2 + (y - 1)^2 - (z - 1)^2 = A^2 \times [ (x - 1)\div 1 ]^2 + [ (y - 1)\div 1 ]^2 + [ (z - 1)\div -1 ]^2\\(x - 1)^2 + (y - 1)^2 - (z - 1)^2 = A^2 \times (x - 1)^2 + (y - 1)^2 + (z - 1)^2$

Since the radius of the cone is not specified, we cannot determine the exact value of A. However, we can represent it as R for now.

$(x - 1)^2 + (y - 1)^2 - (z - 1)^2 = R^2 \times (x - 1)^2 + (y - 1)^2 + (z - 1)^2$

Simplifying the equation, we have:

$(1 - R^2) \times (x - 1)^2 + (1 - R^2) \times (y - 1)^2 - (1 + R^2) \times (z - 1)^2 = 0$

This equation represents the cone with the vertex at (1, 1, 1) and the guiding curve $x^2 + y^2 + z^2 = 1$  and $x + y + z = 1$. The value of R represents the radius of the cone, which is unknown based on the given information.

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