Find parametric equations for the tangent line to the curve of intersection of the paraboloid
z = x^2 + y^2
and the ellipsoid
6x^2 + 5y^2 + 7z^2 = 39
at the point
(−1, 1, 2)
testing
To find the parametric equations for the tangent line to the curve of intersection of the paraboloid and the ellipsoid, first, we need to find the equation of the curve at the given point (-1, 1, 2).
1. Find the partial derivatives of both equations with respect to x, y, and z.
Partial derivative of z = x^2 + y^2 with respect to x:
∂z/∂x = 2x
Partial derivative of z = x^2 + y^2 with respect to y:
∂z/∂y = 2y
Partial derivative of 6x^2 + 5y^2 + 7z^2 = 39 with respect to x:
∂(6x^2 + 5y^2 + 7z^2)/∂x = 12x
Partial derivative of 6x^2 + 5y^2 + 7z^2 = 39 with respect to y:
∂(6x^2 + 5y^2 + 7z^2)/∂y = 10y
Partial derivative of 6x^2 + 5y^2 + 7z^2 = 39 with respect to z:
∂(6x^2 + 5y^2 + 7z^2)/∂z = 14z
2. Plug in the coordinates of the given point (-1, 1, 2) into the partial derivatives to find the slope of the tangent line.
Evaluating the partial derivatives at (-1, 1, 2):
∂z/∂x = 2(-1) = -2
∂z/∂y = 2(1) = 2
∂(6x^2 + 5y^2 + 7z^2)/∂x = 12(-1) = -12
∂(6x^2 + 5y^2 + 7z^2)/∂y = 10(1) = 10
∂(6x^2 + 5y^2 + 7z^2)/∂z = 14(2) = 28
The slope of the tangent line is given by the ratios of the partial derivatives:
m = ∂z/∂x / ∂(6x^2 + 5y^2 + 7z^2)/∂x = -2 / -12 = 1/6
n = ∂z/∂y / ∂(6x^2 + 5y^2 + 7z^2)/∂y = 2 / 10 = 1/5
o = ∂z/∂z / ∂(6x^2 + 5y^2 + 7z^2)/∂z = 28 / 14 = 2
3. The parametric equations for the tangent line can be written as:
x = -1 + at
y = 1 + bt
z = 2 + ct
where a, b, and c are the direction ratios in the x, y, and z directions respectively.
4. Substitute the slope ratios into the parametric equations to find the direction ratios.
Using the given point (-1, 1, 2), substitute the values of x, y, and z in the parametric equations:
-1 = -1 + a * t
1 = 1 + b * t
2 = 2 + c * t
Simplifying the equations, we find:
a * t = 0
b * t = 0
c * t = 0
Since t ≠ 0 for nontrivial solutions, a = b = c = 0. Therefore, the direction ratios of the tangent line are 0, 0, and 0.
Hence, the parametric equations for the tangent line to the curve of intersection of the paraboloid and the ellipsoid at the point (-1, 1, 2) are:
x = -1
y = 1
z = 2
To find the parametric equations for the tangent line to the curve of intersection of the paraboloid and the ellipsoid, we need to follow these steps:
Step 1: Find the equation of the curve of intersection
Step 2: Find the partial derivatives of the two surfaces
Step 3: Calculate the gradient vectors of the partial derivatives
Step 4: Determine the direction vector of the tangent line
Step 5: Write the parametric equations for the tangent line
Let's go through each step in detail:
Step 1: Find the equation of the curve of intersection
To find the curve of intersection, we set the equations of the paraboloid and ellipsoid equal to each other:
x^2 + y^2 = 6x^2 + 5y^2 + 7z^2 - 39
Simplifying this equation gives:
5x^2 + 4y^2 - 7z^2 = -39
Step 2: Find the partial derivatives of the two surfaces
The partial derivatives of the paraboloid are:
∂z/∂x = 2x
∂z/∂y = 2y
The partial derivatives of the ellipsoid are:
∂x/∂x = 10x
∂y/∂y = 8y
∂z/∂z = 14z
Step 3: Calculate the gradient vectors of the partial derivatives
The gradient vector of the paraboloid is given by:
∇f = (∂z/∂x, ∂z/∂y) = (2x, 2y)
The gradient vector of the ellipsoid is given by:
∇g = (∂x/∂x, ∂y/∂y, ∂z/∂z) = (10x, 8y, 14z)
Step 4: Determine the direction vector of the tangent line
To find the direction vector of the tangent line, we need to find the cross product of the gradient vectors:
∇f × ∇g = (2x, 2y, 0) × (10x, 8y, 14z)
= (28yz, -28xz, 20xy - 20yz)
Step 5: Write the parametric equations for the tangent line
Now that we have the direction vector, we can write the parametric equations for the tangent line.
Let the point of tangency be (x0, y0, z0). Then the equation of the tangent line is given by:
x = x0 + t(28yz0)
y = y0 - t(28xz0)
z = z0 + t(20xy0 - 20yz0)
Given that the point of tangency is (-1, 1, 2), we can substitute these values into the parametric equations to get the final answer.