If the centre of a circle is `(3, 4)` and its size is just sufficient to contain to circle `x^(2) + y^(2) =1`, then the radius of the required circle is _____.

Answer

To find the radius of the required circle that contains the circle defined by the equation \(x^2 + y^2 = 1\) and has its center at the point \((3, 4)\), we can follow these steps:

### Step 1: Identify the properties of the smaller circle
The equation \(x^2 + y^2 = 1\) represents a circle centered at the origin \((0, 0)\) with a radius of \(1\).

### Step 2: Determine the center of the required circle
The center of the required circle is given as \((3, 4)\).

### Step 3: Calculate the distance from the center of the required circle to the center of the smaller circle
We will use the distance formula to find the distance \(OP\) between the center of the smaller circle \(O(0, 0)\) and the center of the required circle \(P(3, 4)\). The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ OP = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Calculate the radius of the required circle
The radius \(R\) of the required circle must be the distance \(OP\) plus the radius of the smaller circle. Since the radius of the smaller circle is \(1\), we have:
\[ R = OP + 1 = 5 + 1 = 6 \]

### Conclusion
The radius of the required circle is \(6\).