Difference in values of the radius of a circle whose center is at the origin and which touches the circle `x^2+y^2-6x-8y+21=0` is_____________

Explanation

To solve the problem, we need to find the difference in the values of the radius of a circle whose center is at the origin (0, 0) and which touches the given circle defined by the equation \(x^2 + y^2 - 6x - 8y + 21 = 0\).

### Step-by-Step Solution:

1. **Rewrite the given circle equation**:
The given circle equation is \(x^2 + y^2 - 6x - 8y + 21 = 0\). We can rewrite it in standard form by completing the square.

\[ x^2 - 6x + y^2 - 8y + 21 = 0 \]

Completing the square for \(x\):
\[ x^2 - 6x = (x - 3)^2 - 9 \]

Completing the square for \(y\):
\[ y^2 - 8y = (y - 4)^2 - 16 \]

Substituting back:
\[ (x - 3)^2 - 9 + (y - 4)^2 - 16 + 21 = 0 \]

Simplifying:
\[ (x - 3)^2 + (y - 4)^2 - 4 = 0 \]
\[ (x - 3)^2 + (y - 4)^2 = 4 \]

This shows that the center of the given circle is at \((3, 4)\) and its radius is \(2\) (since \(\sqrt{4} = 2\)).

2. **Calculate the distance from the origin to the center of the given circle**:
The distance \(d\) from the origin \((0, 0)\) to the center \((3, 4)\) is calculated using the distance formula:
\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Set up the equation for the radius of the circle centered at the origin**:
Let the radius of the circle centered at the origin be \(r\). Since the circle touches the given circle, the relationship between the radii and the distance from the origin to the center of the given circle is:
\[ r + 2 = 5 \quad \text{(since the distance from the origin to the center is 5)} \]

Rearranging gives:
\[ r = 5 - 2 = 3 \]

4. **Consider the other case**:
The other case is when the circle centered at the origin is inside the given circle:
\[ 5 - r = 2 \]
Rearranging gives:
\[ r = 5 - 2 = 3 \]

Thus, we have two possible values for \(r\):
- \(r_1 = 3\) (when the circle is outside)
- \(r_2 = 7\) (when the circle is inside)

5. **Calculate the difference in radii**:
The difference in the values of the radius is:
\[ |r_2 - r_1| = |7 - 3| = 4 \]

### Final Answer:
The difference in values of the radius of the circle is **4 units**.