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

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In summary, (x,y,z)∈R^3 are points that lie on the plane x+2y+3z=78 and the sphere x^2+y^2+z^2=468. The intersection of a plane and a sphere is a circle, and finding the equation of this circle involves solving the equations simultaneously. The maximum value of x can be found by finding the radius of this circle, but there are multiple approaches to solving this problem. The value of a+b can then be calculated once the maximum value of x is known.
  • #1
jingu
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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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  • #2
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.
 
  • #3
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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  • #4
So can anyone give me its complete solution,?

so that I can understand the concept.
 
  • #5
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 [itex]x^2+y^2+z^2=468[/itex] 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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  • #6
I think it would be a circle.Please check whether I am correct or not...
 
  • #8
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?
 
  • #9
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!
 

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

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