hitachiin69 said:Find the radius of smallest sphere that is tangent to both the lines
L1 :x=t+1,y=2t+4,z=−3t+5 and L2 :x=4t−12,y=t+5,z=t+17.
Plato said:This is a busy-work problem.
Although I have not done the basic algebra, it appears that those two lines are skew lines. (you may need to show that)
Two skew lines share a unique perpendicular. Its length is the diameter of the sphere.
Now, one does not need to find that perpendicular. Just find the distance between the two lines.
Your text ought to discuss that somewhere. It may be in a problem set.
The radius of a sphere tangent to two lines is the distance from the center of the sphere to the point of tangency on one of the lines. It is also equal to the distance from the center to the other line, since the sphere is tangent to both lines.
The radius can be calculated using the Pythagorean theorem: r = √(d1^2 + d2^2), where d1 and d2 are the distances from the center of the sphere to each of the lines.
No, the radius of a sphere cannot be negative. It represents a distance and therefore must be a positive value.
The angles formed by the two lines and the radius of the sphere are directly related. Specifically, the angles formed by the lines at the point of tangency are equal, and they are also equal to the angle between the lines and the radius of the sphere.
Yes, this concept is commonly used in geometry and engineering, particularly in designing structures such as bridges and tunnels. It can also be used in various geometric constructions and calculations.