What Is the Maximum Value of x for Points on a Plane and Sphere in 3D Geometry?

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The discussion revolves around finding the maximum value of x for points that lie on both a specified plane and a sphere in 3D geometry. Participants explore the intersection of the plane and sphere, concluding that it forms a circle. They suggest solving the equations simultaneously to express two variables in terms of the third, which is essential for determining the maximum x value. There is also a focus on calculating the radius of the circle formed by the intersection, with some participants seeking hints and guidance on the next steps. Overall, the conversation emphasizes understanding the geometric relationship between the plane and sphere to solve the problem effectively.
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(x,y,z)∈R^3 are points that lie on the plane x+2y+3z=78, and lie on the sphere x^2+y^2+z^2=468. The maximum value of x has the form a/b, where a and b are coprime positive integers. What is the value of a+b?
 
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Here is a hint: How does the intersection of a plane and a sphere look like?
Once you know the maximum value of x, calculating a+b should be easy.
 
hello friend, i even don't know from which topic is this question,help?

mfb said:
Here is a hint: How does the intersection of a plane and a sphere look like?
Once you know the maximum value of x, calculating a+b should be easy.
i even don't know from which topic is this question,please help?
 
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So can anyone give me its complete solution,?

so that I can understand the concept.
 
Can you first answer mfb's question? What does the intersection of a sphere and a plane look like? What kind of figure is that?

You can find the equation of that graph by solving the two equations, x+2y+3z=78, and x^2+y^2+z^2=468 simultaneously. Since that is two equations in three variables, you can solve for two, say x and y, in terms of the third.
 
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I think it would be a circle.Please check whether I am correct or not...
 
It is a circle, right.
 
then what to do, you guys just tell me the steps I will do all by my own,so what will be the next step?
 
mfb said:
It is a circle, right.
then i think we have to find the radius of this circle...am i correct...?
 
  • #10
That is possible. I used a different approach, but there are many ways to solve this.

Can you link the source of the question? If it is not a current question, I might give more hints.
 
  • #11
mfb said:
That is possible. I used a different approach, but there are many ways to solve this.

Can you link the source of the question? If it is not a current question, I might give more hints.

yes give me hints...
 
  • #12
I think the radius is 5.78, and that is not the answer...help!
 

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