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

[jingu]

(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?

 

[mfb]

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.

 

[jingu]

hello friend, i even don't know from which topic is this question,help?

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?

 

[jingu]

So can anyone give me its complete solution,?

so that I can understand the concept.

 

[HallsofIvy]

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.

 

[jingu]

I think it would be a circle.Please check whether I am correct or not...

 

[mfb]

It is a circle, right.

 

[jingu]

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?

 

[jingu]

It is a circle, right.

then i think we have to find the radius of this circle...am i correct...?

 

[mfb]

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.

 

[jingu]

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...

 

[jingu]

I think the radius is 5.78, and that is not the answer...help!