What is the equation for the circle formed by the intersection of two spheres?

In summary: A and B is greater than 5, and the sphere of intersection is given by the simultaneous solution to the equation of either of the spheres and the equation of that plane.
  • #1
songoku
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319

Homework Statement


In the space, consider the sphere S1 of radius 3 whose center is the point A (1, -1, 1) and the sphere S2 of radius 2 whose center is the point B(t, 1 - t, 1 + t).
a. Find the range of values of t in order the two spheres S1 and S2 have common points
b. Find the value of t for which B is closest to the point A
c. For the value of B obtained from (b), find the radius of circle formed as intersection of S1 and S2


Homework Equations


differentiation
equation of sphere: (x - a)2 + (y - b)2 + (z - c)2 = r2


The Attempt at a Solution


I tried to equate the two equations of sphere and set discriminant ≥ 0, but I ended up having equation containing three variables x, y and z which can't be solved.

To be honest, the only equation I know about sphere is the one I wrote above...:redface:
 
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  • #2
It's not really all that much about the equation of the spheres. If the distance between A and B is greater then 5, then the spheres don't intersect. If it's less than 5 then they do. It's about the distance between A and B.
 
  • #3
Dick said:
It's not really all that much about the equation of the spheres. If the distance between A and B is greater then 5, then the spheres don't intersect. If it's less than 5 then they do. It's about the distance between A and B.

Ah, why don't I think about it...:blushing:

OK let's move on to find part (c). Actually I don't know which circle the question is referring to. I have image in my head that the intersection between two spheres are another 3 dimensional object, not a 2 dimensional object such as circle...
 
  • #4
Go ahead and multiply out the two equations for the circles, then subtract one equation from the other. Since all "square" terms have coefficient 1, they will cancel giving a linear equation- the equation of the plane in which the two spheres intersect.

Then the circle of intersection is given by the simultaneous solution to the equation of either of the spheres and the equation of that plane.
 
  • #5
HallsofIvy said:
Go ahead and multiply out the two equations for the circles, then subtract one equation from the other. Since all "square" terms have coefficient 1, they will cancel giving a linear equation- the equation of the plane in which the two spheres intersect.

Then the circle of intersection is given by the simultaneous solution to the equation of either of the spheres and the equation of that plane.

I found the equation of plane: 2y + 2z = 7

I can't find the simultaneous solution to the equation of sphere 1 and the plane. I substitute y = 7/2 - z to the equation of plane and end up having two variables, x and z? What should I do?

Thanks
 

1. What is the intersection of two spheres?

The intersection of two spheres is the set of points where the two spheres overlap or intersect with each other. It can be visualized as the common region between the two spheres.

2. How many points are in the intersection of two spheres?

The number of points in the intersection of two spheres can vary. It can be zero, one, or multiple points depending on the positions and sizes of the two spheres.

3. Can two spheres intersect at more than one point?

Yes, it is possible for two spheres to intersect at more than one point. This occurs when the two spheres have a common tangent plane, resulting in two points of intersection.

4. How can the intersection of two spheres be calculated?

The intersection of two spheres can be calculated by using the distance formula to find the distance between the centers of the spheres. If this distance is less than the sum of the radii of the spheres, then they intersect. The equation of the intersection can also be found by solving the system of equations for the two spheres.

5. What are the real-life applications of the intersection of two spheres?

The intersection of two spheres has many real-life applications in fields such as physics, engineering, and computer graphics. It is used in collision detection algorithms for objects in motion, calculating the overlap of atoms in molecules, and creating 3D models in computer graphics. It can also be used in astronomy to determine the position of celestial objects in the sky.

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