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

### Step 1: Define the Circles
Let the equations of the two given circles be:
1. Circle 1: \((x - a_1)^2 + (y - b_1)^2 = r_1^2\)
2. Circle 2: \((x - a_2)^2 + (y - b_2)^2 = r_2^2\)

### Step 2: General Equation of the Family of Circles
The general equation of a circle can be expressed as:
\[ (x - c)^2 + (y - d)^2 = r^2 \]
where \(c\), \(d\), and \(r\) are parameters that define the circle.

### Step 3: Set Up the Condition for External Tangency
For the family of circles to touch the two given circles externally, the distance between the centers of the circles must equal the sum of their radii. This gives us two equations based on the tangency condition:
1. For Circle 1:
\[ \sqrt{(c - a_1)^2 + (d - b_1)^2} = r + r_1 \]
2. For Circle 2:
\[ \sqrt{(c - a_2)^2 + (d - b_2)^2} = r + r_2 \]

### Step 4: Square Both Sides to Eliminate the Square Root
Squaring both sides of the equations gives:
1. \((c - a_1)^2 + (d - b_1)^2 = (r + r_1)^2\)
2. \((c - a_2)^2 + (d - b_2)^2 = (r + r_2)^2\)

### Step 5: Rearranging the Equations
Rearranging the equations leads to:
1. \((c - a_1)^2 + (d - b_1)^2 - (r + r_1)^2 = 0\)
2. \((c - a_2)^2 + (d - b_2)^2 - (r + r_2)^2 = 0\)

### Step 6: Eliminate the Radius \(r\)
To find a relationship between \(c\), \(d\), and the parameters of the given circles, we can eliminate \(r\) from these equations. This will typically involve substituting one equation into the other.

### Step 7: Determine the Number of Unknowns
After eliminating \(r\), we will have an equation in terms of \(c\) and \(d\). The number of unknowns in this equation will determine the order of the differential equation.

### Step 8: Conclusion
In this case, since \(a_1\), \(b_1\), \(r_1\), \(a_2\), \(b_2\), and \(r_2\) are known constants, the only unknowns are \(c\) and \(d\). Thus, we have two unknowns.

The order of the differential equation is equal to the number of unknown parameters we need to eliminate. Since we have two unknowns, the order of the differential equation of the family of circles touching the two given circles externally is **2**.

### Final Answer
The order of the differential equation of the family of circles touching two given circles externally is **2**.

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