The order of the differential equation of family of circles touching two given circles externally is

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To find the order of the differential equation of the family of circles touching two given circles externally, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to determine the order of the differential equation representing the family of circles that touch two given circles externally.

2. **Setting Up the Circles**: Let the equations of the two given circles be:
- Circle 1: \( (x - a)^2 + (y - b)^2 = r_1^2 \)
- Circle 2: \( (x - c)^2 + (y - d)^2 = r_2^2 \)

Here, \( (a, b) \) and \( (c, d) \) are the centers of the circles, and \( r_1 \) and \( r_2 \) are their respective radii.

3. **Condition for External Tangency**: For a circle with center \( (h, k) \) and radius \( r \) to touch the two given circles externally, the following conditions must be satisfied:
- The distance from \( (h, k) \) to the center of Circle 1 must equal \( r + r_1 \).
- The distance from \( (h, k) \) to the center of Circle 2 must equal \( r + r_2 \).

This gives us two equations:
\[ \sqrt{(h - a)^2 + (k - b)^2} = r + r_1 \]
\[ \sqrt{(h - c)^2 + (k - d)^2} = r + r_2 \]

4. **Squaring Both Sides**: To eliminate the square roots, we square both sides of the equations:
\[ (h - a)^2 + (k - b)^2 = (r + r_1)^2 \]
\[ (h - c)^2 + (k - d)^2 = (r + r_2)^2 \]

5. **Expanding the Equations**: Expanding both equations gives:
- From the first equation:
\[ (h - a)^2 + (k - b)^2 = r^2 + 2rr_1 + r_1^2 \]
- From the second equation:
\[ (h - c)^2 + (k - d)^2 = r^2 + 2rr_2 + r_2^2 \]

6. **Eliminating \( r \)**: We can eliminate \( r \) from these equations to find a relationship between \( h, k, a, b, c, d, r_1, \) and \( r_2 \).

7. **Finding the Order of the Differential Equation**: The order of the differential equation corresponds to the number of arbitrary constants in the family of curves. In this case, the family of circles can be described by the parameters \( h, k, \) and \( r \).

Since we have one arbitrary radius \( r \) and two centers \( (h, k) \), we can conclude that the family of circles is defined by one parameter. Therefore, the order of the differential equation is **1**.

### Final Answer:
The order of the differential equation of the family of circles touching two given circles externally is **1**.