If a circle whose center is `(1,-3)` touches the line `3x-4y-5=0` , then find its radius.

Explanation

To find the radius of a circle whose center is at the point (1, -3) and that touches the line given by the equation \(3x - 4y - 5 = 0\), we will follow these steps:

### Step 1: Identify the center of the circle and the line equation
The center of the circle is given as \(C(1, -3)\) and the line equation is \(3x - 4y - 5 = 0\).

### Step 2: Use the formula for the distance from a point to a line
The distance \(d\) from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
In our case, \(A = 3\), \(B = -4\), \(C = -5\), and the point is \((x_1, y_1) = (1, -3)\).

### Step 3: Substitute the values into the distance formula
Substituting the values into the formula:
\[ d = \frac{|3(1) + (-4)(-3) - 5|}{\sqrt{3^2 + (-4)^2}} \]

### Step 4: Calculate the numerator
Calculating the numerator:
\[ 3(1) + (-4)(-3) - 5 = 3 + 12 - 5 = 10 \]
Thus, the absolute value is \(|10| = 10\).

### Step 5: Calculate the denominator
Calculating the denominator:
\[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 6: Calculate the distance
Now substituting back into the distance formula:
\[ d = \frac{10}{5} = 2 \]

### Conclusion
Since the distance \(d\) we calculated is the radius of the circle, the radius of the circle is \(2\) units.

### Final Answer
The radius of the circle is \(2\) units.

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