there are two circles touching each other at one point. The radius of bigger circle is 4cm and smaller is 1cm. A small third circle is drawn which touches both the first two circles at one point each. A common tangent passes through all the three circles.Find the radius of this third cirlcle.

Question

there are two circles touching each other at one point. The radius of bigger circle is 4cm and smaller is 1cm. A small third circle is drawn which touches both the first two circles at one point each. A common tangent passes through all the three circles.Find the radius of this third cirlcle.

Answer 1

This problem is discussed at

http://mathworld.wolfram.com/TangentCircles.html

starting at equation (17), where we see that

1/√r = 1/√4 + 1/√1
1/√r = 1/2 + 1
1/√r = 3/2
r = 4/9

Answer 2

To find the radius of the third circle, we can use the concept of tangents and common tangents.

Let's call the center of the larger circle O1, the center of the smaller circle O2, and the center of the third circle O3.

From the given information, we know that the radius of the larger circle (r1) is 4 cm and the radius of the smaller circle (r2) is 1 cm. The two circles touch each other at one point, which means the distance between the two centers is equal to the sum of their radii, i.e., O1O2 = r1 + r2.

The common tangent of two circles is perpendicular to the line joining their centers at the point of tangency. As the small third circle is tangent to both the larger and smaller circles, the tangent line to all three circles will pass through the points of tangency.

Now, let's consider the line joining the centers of the larger and smaller circles (O1O2). The point of tangency of the third circle with the larger circle lies on this line. Let's call this point T. Similarly, the point of tangency with the smaller circle is called S.

To find the radius of the third circle, we need to find the distance between O3 and the line O1O2.

Here's how we can find the radius of the third circle:

1. Find the distance O1O2:
O1O2 = r1 + r2 = 4 cm + 1 cm = 5 cm.

2. Find the distance O1T (or O2T) using the property of tangents:
O1T = O2T = sqrt((O1O2)^2 - (r1 - r2)^2)
O1T = O2T = sqrt((5 cm)^2 - (4 cm - 1 cm)^2)
O1T = O2T = sqrt(25 cm^2 - 9 cm^2)
O1T = O2T = sqrt(16 cm^2)
O1T = O2T = 4 cm.

3. Find the distance between the line O1O2 and O3:
The distance between O1O2 and O3 is equal to the radius of the third circle. Let's call this r3.

4. The distance between O1T (or O2T) and O3 is equal to the sum of the radius of the third circle and the radius of the smaller circle, i.e., r3 + r2.
Since O1T (or O2T) = 4 cm, we have:
4 cm = r3 + 1 cm.
r3 = 3 cm.

Therefore, the radius of the third circle is 3 cm.