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Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,-1,1)?
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Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,...
Introduction
To find the points on the sphere x^2 + y^2 + z^2 = 4 that are farthest from the point (1, -1, 1), we can use the concept of distance between points in three-dimensional space. By finding the points that have the maximum distance from the given point, we can determine the farthest points on the sphere.
Steps to Find the Farthest Points
- Convert the given equation of the sphere into standard form.
- Apply the distance formula to find the distance between a point on the sphere and the given point.
- Maximize the distance function subject to the constraint of the sphere equation using Lagrange multipliers.
- Find the critical points of the distance function.
- Determine which critical points satisfy the sphere equation.
- Calculate the distance from each valid critical point to the given point.
- Identify the point(s) with the maximum distance as the farthest point(s) on the sphere.
Detailed Explanation
1. Convert the equation to standard form
The given equation of the sphere is x^2 + y^2 + z^2 = 4. This can be rearranged to the standard form as
(x - 0)^2 + (y - 0)^2 + (z - 0)^2 = 2^2.
Therefore, the center of the sphere is at the origin (0, 0, 0) and the radius is 2.
2. Apply the distance formula
The distance between two points (x1, y1, z1) and (x2, y2, z2) can be calculated using the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2].
In this case, we want to find the distance between a point on the sphere (x, y, z) and the given point (1, -1, 1). So, the distance function becomes:
d = √[(x - 1)^2 + (y + 1)^2 + (z - 1)^2].
3. Maximize the distance function
To find the farthest points on the sphere, we need to maximize the distance function while satisfying the constraint of the sphere equation. We can use Lagrange multipliers for this optimization problem.
Let λ be the Lagrange multiplier. The function to maximize becomes:
F(x, y, z, λ) = (x - 1)^2 + (y + 1)^2 + (z - 1)^2 + λ(x^2 + y^2 + z^2 - 4).
4. Find the critical points
To find the critical points, we take the partial derivatives of F with respect to x, y, z, and λ, and set them equal to zero:
∂F/∂x = 2(x - 1) + 2λx = 0,
∂F/∂y
Community Answer
Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,...
Introduction:
We are given the equation of a sphere, x^2 + y^2 + z^2 = 4, and we need to find the points on the sphere that are farthest from the point (1, -1, 1). To solve this problem, we will use the concept of distance formula and calculus.
Step 1: Find the distance between a point on the sphere and the given point:
Let's consider a general point (x, y, z) on the sphere. The distance between this point and the given point (1, -1, 1) can be calculated using the distance formula:
d = √((x - 1)^2 + (y + 1)^2 + (z - 1)^2)
Step 2: Maximize the distance function:
To find the points on the sphere that are farthest from the given point, we need to maximize the distance function d. We can do this by taking partial derivatives of d with respect to x, y, and z, and setting them equal to zero:
∂d/∂x = 0
∂d/∂y = 0
∂d/∂z = 0
Step 3: Solve the partial derivative equations:
By solving the partial derivative equations, we can find the critical points on the sphere where the distance function is maximized. To solve these equations, we differentiate d with respect to each variable separately:
∂d/∂x = (x - 1) / √((x - 1)^2 + (y + 1)^2 + (z - 1)^2) = 0
∂d/∂y = (y + 1) / √((x - 1)^2 + (y + 1)^2 + (z - 1)^2) = 0
∂d/∂z = (z - 1) / √((x - 1)^2 + (y + 1)^2 + (z - 1)^2) = 0
Step 4: Solve for the critical points:
Solving the partial derivative equations will give us the critical points on the sphere where the distance function is maximized. By solving these equations, we find that the critical points are:
x = 1, y = -1, z = 1
Step 5: Determine the farthest points:
Now that we have the critical points, we need to determine which of these points are farthest from the given point. To do this, we substitute the critical points into the distance function d and calculate the distances:
d1 = √((1 - 1)^2 + (-1 + 1)^2 + (1 - 1)^2) = 0
d2 = √((1 - 1)^2 + (-1 + 1)^2 + (1 - 1)^2) = 0
Conclusion:
The critical points on the sphere that are farthest from the given point (1, -1, 1) are (1, -1, 1). These points have a distance of 0 from the given point, indicating that they are the farthest points
We are given the equation of a sphere, x^2 + y^2 + z^2 = 4, and we need to find the points on the sphere that are farthest from the point (1, -1, 1). To solve this problem, we will use the concept of distance formula and calculus.
Step 1: Find the distance between a point on the sphere and the given point:
Let's consider a general point (x, y, z) on the sphere. The distance between this point and the given point (1, -1, 1) can be calculated using the distance formula:
d = √((x - 1)^2 + (y + 1)^2 + (z - 1)^2)
Step 2: Maximize the distance function:
To find the points on the sphere that are farthest from the given point, we need to maximize the distance function d. We can do this by taking partial derivatives of d with respect to x, y, and z, and setting them equal to zero:
∂d/∂x = 0
∂d/∂y = 0
∂d/∂z = 0
Step 3: Solve the partial derivative equations:
By solving the partial derivative equations, we can find the critical points on the sphere where the distance function is maximized. To solve these equations, we differentiate d with respect to each variable separately:
∂d/∂x = (x - 1) / √((x - 1)^2 + (y + 1)^2 + (z - 1)^2) = 0
∂d/∂y = (y + 1) / √((x - 1)^2 + (y + 1)^2 + (z - 1)^2) = 0
∂d/∂z = (z - 1) / √((x - 1)^2 + (y + 1)^2 + (z - 1)^2) = 0
Step 4: Solve for the critical points:
Solving the partial derivative equations will give us the critical points on the sphere where the distance function is maximized. By solving these equations, we find that the critical points are:
x = 1, y = -1, z = 1
Step 5: Determine the farthest points:
Now that we have the critical points, we need to determine which of these points are farthest from the given point. To do this, we substitute the critical points into the distance function d and calculate the distances:
d1 = √((1 - 1)^2 + (-1 + 1)^2 + (1 - 1)^2) = 0
d2 = √((1 - 1)^2 + (-1 + 1)^2 + (1 - 1)^2) = 0
Conclusion:
The critical points on the sphere that are farthest from the given point (1, -1, 1) are (1, -1, 1). These points have a distance of 0 from the given point, indicating that they are the farthest points
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Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,-1,1)?
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Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,-1,1)? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,-1,1)? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,-1,1)?.
Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,-1,1)? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,-1,1)? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the points at the sphere x^2 y^2 z^2=4, farest from the point (1,-1,1)?.
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