If the lines `5x+12y=3` and `10x+ 24y-58=0` are tangents to a circle, then find the radius of the circle.

Answer

To find the radius of the circle given that the lines \(5x + 12y = 3\) and \(10x + 24y - 58 = 0\) are tangents to the circle, we can follow these steps:

### Step 1: Rewrite the second line equation
The second line equation is \(10x + 24y - 58 = 0\). We can simplify this by dividing the entire equation by 2:

\[ 5x + 12y - 29 = 0 \]

### Step 2: Identify the equations of the lines
Now we have two equations:
1. \(5x + 12y = 3\) (let's call this Line 1)
2. \(5x + 12y = 29\) (let's call this Line 2)

### Step 3: Check if the lines are parallel
Both lines have the same coefficients for \(x\) and \(y\), which means they are parallel. The distance between two parallel lines can be calculated using the formula:

\[ \text{Distance} = \frac{|c_2 - c_1|}{\sqrt{a^2 + b^2}} \]

where \(c_1\) and \(c_2\) are the constants from the line equations \(Ax + By = C\).

### Step 4: Identify constants
From Line 1, we have:
- \(c_1 = 3\)

From Line 2, we have:
- \(c_2 = 29\)

The coefficients \(A\) and \(B\) are:
- \(A = 5\)
- \(B = 12\)

### Step 5: Calculate the distance between the lines
Substituting the values into the distance formula:

\[ \text{Distance} = \frac{|29 - 3|}{\sqrt{5^2 + 12^2}} = \frac{26}{\sqrt{25 + 144}} = \frac{26}{\sqrt{169}} = \frac{26}{13} = 2 \]

### Step 6: Find the radius of the circle
Since the distance between the two tangents is equal to the diameter of the circle, the radius \(r\) is half of the distance:

\[ r = \frac{2}{2} = 1 \]

### Final Answer
The radius of the circle is \(1\) unit.

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