Minimum and Maximum Distances with Circles
Lesson · Intermediate
Geometry: Plane Geometry
Point P and the circle are fixed. A lies on the circle and can move.
The two circles are fixed. Points P and Q lie on the circles ω₁ and ω₂, respectively, and can move.
By Triangle Inequality in triangle APO, this is correct.
Equality holds when P, A, and O are collinear, so A = B.
By Triangle Inequality in triangle APO, this is correct.
Equality holds when P, A, and O are collinear, so A = C.
In O₁PQO₂
Equality holds when Q = D and P = A.
In O₁PQO₂
Equality holds when Q, O₁, and O₂ are collinear, so Q = C and P = B.
A circle has radius 13, and a fixed point P lies inside the circle. As A moves around the circle, the maximum value of PA is 18. Find the minimum value of PA.
Two circles have radii 7 and 10. The maximum possible distance between a point on one circle and a point on the other is 25. Determine whether the circles intersect.
Let the center distance be d.
Since the maximum distance is
we get
Now compare:
and
Since
the circles intersect at two points.
Two circles have radii R and r, with R > r. The smaller circle lies entirely inside the larger circle without touching it. The minimum and maximum possible distances between points on the two circles are 4 and 28, respectively. If the distance between the centers is 6, find R and r.
Thus
Solve the system:
Adding,
so
Then
Therefore,