There are three circles,with radii 1 cm, 2 cm and 3 cm, tangent to each other. Find the radius of the fourth circle, which is tangent to all the existing circles

Options

A. root6 cm
B. 2
C. 6/2
D. BOTH (A) AND ( C )

Answer

To find the radius of the fourth circle that is tangent to three existing circles with radii 1 cm, 2 cm, and 3 cm, we can use a formula derived from Descartes' Circle Theorem.

### Step-by-Step Solution:

**Step 1: Identify the radii of the existing circles.**
- Let \( r_1 = 1 \) cm (radius of the first circle)
- Let \( r_2 = 2 \) cm (radius of the second circle)
- Let \( r_3 = 3 \) cm (radius of the third circle)

**Step 2: Use the formula for the radius of the fourth circle.**
According to Descartes' Circle Theorem, if four circles are tangent to each other, the following relationship holds for their curvatures (the curvature \( k \) of a circle is defined as \( k = \frac{1}{r} \)):
\[ (k_1 + k_2 + k_3 + k_4)^2 = 2(k_1^2 + k_2^2 + k_3^2 + k_4^2) \]
Where \( k_4 \) is the curvature of the fourth circle.

**Step 3: Calculate the curvatures of the existing circles.**
- The curvature of the first circle \( k_1 = \frac{1}{r_1} = \frac{1}{1} = 1 \)
- The curvature of the second circle \( k_2 = \frac{1}{r_2} = \frac{1}{2} = 0.5 \)
- The curvature of the third circle \( k_3 = \frac{1}{r_3} = \frac{1}{3} \approx 0.333 \)

**Step 4: Substitute the curvatures into the formula.**
Let \( k_4 \) be the curvature of the fourth circle. The equation becomes:
\[ (1 + 0.5 + 0.333 + k_4)^2 = 2(1^2 + 0.5^2 + (0.333)^2 + k_4^2) \]

**Step 5: Simplify and solve for \( k_4 \).**
Calculating the left side:
\[ (1 + 0.5 + 0.333 + k_4) = (1.833 + k_4) \]
Calculating the right side:
\[ 2(1 + 0.25 + 0.111 + k_4^2) = 2(1.361 + k_4^2) = 2.722 + 2k_4^2 \]

Now we set the two sides equal:
\[ (1.833 + k_4)^2 = 2.722 + 2k_4^2 \]

**Step 6: Expand and rearrange the equation.**
Expanding the left side:
\[ 3.354 + 3.666k_4 + k_4^2 = 2.722 + 2k_4^2 \]
Rearranging gives:
\[ 3.354 - 2.722 + 3.666k_4 - k_4^2 = 0 \]
This simplifies to:
\[ 0.632 + 3.666k_4 - k_4^2 = 0 \]

**Step 7: Solve the quadratic equation for \( k_4 \).**
Rearranging gives:
\[ k_4^2 - 3.666k_4 - 0.632 = 0 \]
Using the quadratic formula \( k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ k_4 = \frac{3.666 \pm \sqrt{(3.666)^2 + 4 \cdot 0.632}}{2} \]

**Step 8: Calculate \( k_4 \) and find the radius \( r_4 \).**
After calculating \( k_4 \), we find the radius \( r_4 = \frac{1}{k_4} \).

### Final Result:
The radius of the fourth circle is approximately \( \sqrt{6} \) cm.