The number of common tangents to `x^(2)+y^(2)=256, (x-3)^(2)+(y-4)^(2)=121` is

Options

A. one
B. two
C. four
D. zero

Answer

To find the number of common tangents to the circles given by the equations \(x^2 + y^2 = 256\) and \((x-3)^2 + (y-4)^2 = 121\), we can follow these steps:

### Step 1: Identify the Radii of the Circles
The first circle is given by the equation:
\[ x^2 + y^2 = 256 \]
This can be rewritten as:
\[ x^2 + y^2 = 16^2 \]
Thus, the radius \(R_1\) of the first circle is:
\[ R_1 = 16 \]

The second circle is given by the equation:
\[ (x-3)^2 + (y-4)^2 = 121 \]
This can be rewritten as:
\[ (x-3)^2 + (y-4)^2 = 11^2 \]
Thus, the radius \(R_2\) of the second circle is:
\[ R_2 = 11 \]

### Step 2: Find the Centers of the Circles
The center of the first circle \(C_1\) is at the origin:
\[ C_1 = (0, 0) \]
The center of the second circle \(C_2\) is at:
\[ C_2 = (3, 4) \]

### Step 3: Calculate the Distance Between the Centers
The distance \(d\) between the centers \(C_1\) and \(C_2\) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the centers:
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Compare the Distance with the Sum of the Radii
Now, we need to compare the distance \(d\) with the sum of the radii \(R_1 + R_2\):
\[ R_1 + R_2 = 16 + 11 = 27 \]
We have:
\[ d = 5 \quad \text{and} \quad R_1 + R_2 = 27 \]

### Step 5: Determine the Number of Common Tangents
Since the distance between the centers \(d\) is less than the sum of the radii \(R_1 + R_2\) (i.e., \(5 < 27\)), this indicates that one circle is inside the other without touching it. Therefore, there are no common tangents.

### Final Answer
The number of common tangents to the circles is:
\[ \text{0} \]