DCG short question help!!

Leaving Cert Student

This is from a mock paper its a solids in contact q, I get the first but but lost on bit after that (the second sphere). I am probably wrong but couldn't there be infinitely many spheres satisfying the given conditions?

jimbo28

First off rotate sphere b to to outside of the cone( to the Left).This will appear as a circle in elevation.

Bi-sect the angle between the extreme left generator of the cone and the xy line.

you now need to find the locus of a point that is equidistant from the left generator of the cone and the circle ( sphere b)

where this locus intersects the bi-sector will give you the centre point of the new sphere that is in contact with both the cone and the existing sphere.

Go perpendicular to the left generator of the cone through this new centre point to find the radius.

Now rotate this new sphere into its final position in plan, and project it back up to the elevation of the question.

Leaving Cert Student

I got locus and midpoint in elevation assuming tjats what you meant so i have the sphere in ele but how do i determine its pos. in plan?

koraDCG

You are correct in saying a number of spheres could satisfy these conditions.

To complete this question you must locate 2 construction spheres in elevation;
- Locate a sphere that touches both the inverted cone and the horizontal plane (Construction Sphere 1 or C1)
- Locate a second sphere that touches both sphere B and the horizontal plane. (Construction Sphere 2 or C2)
The two spheres must be equal in radius

1) Locating Centre of C1- touches the cone and the Horizontal Plane
Bi-sect the angle between the extreme left generator of the cone and the xy line
The centre point of our first construction sphere will be on this line

2) Locating Centre of C2- touches the sphere B and the Horizontal Plane
Find the locus of a point that is equidistant from the sphere B and the XY
The centre of our second construction sphere will be on this line

3) As a number of spheres could satisfy these conditions we can choose any height for our centre point (once it is large enough to touch sphere B and the XY). The spheres have the same radius; the centre points are at the same height.

4) Locate both construction spheres in plan
C1 is at the side of the cone
C2 is at the side of the sphere B

5) Rotate the centre of each sphere in plan until the arcs cross.
C1 is at the side of the cone, the centre of the cone is the centre of rotation
C2 is at the side of the sphere B, the centre of B is the centre of rotation

Where the arc cross locates the centre point of sphere C. Solution is below.