If `5x-12y+10=0` and `12y-5x+16=0` are two tangents to a circle, then the radius the circle, is

Options

A. 1
B. 2
C. 4
D. 6

Explanation

To find the radius of the circle given the equations of the tangents, we can follow these steps:

### Step 1: Identify the equations of the tangents
The given equations of the tangents are:
1. \( 5x - 12y + 10 = 0 \) (Equation 1)
2. \( 12y - 5x + 16 = 0 \) (Equation 2)

### Step 2: Rewrite the second equation
We can rearrange Equation 2 to match the form of Equation 1:
\[ -5x + 12y + 16 = 0 \implies 5x - 12y - 16 = 0 \]
Now we have:
1. \( 5x - 12y + 10 = 0 \)
2. \( 5x - 12y - 16 = 0 \)

### Step 3: Determine if the lines are parallel
Since both equations have the same coefficients for \(x\) and \(y\), we can conclude that the lines are parallel.

### Step 4: Calculate the distance between the two parallel lines
The formula for the distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
Here, \(C_1 = 10\) and \(C_2 = -16\), and the coefficients \(A = 5\) and \(B = -12\).

Substituting these values into the formula:
\[ d = \frac{|-16 - 10|}{\sqrt{5^2 + (-12)^2}} = \frac{|-26|}{\sqrt{25 + 144}} = \frac{26}{\sqrt{169}} = \frac{26}{13} = 2 \]

### Step 5: Relate the distance to the radius
Since the distance \(d\) between the two tangents is equal to the diameter of the circle, we have:
\[ d = 2r \]
Thus, we can find the radius \(r\):
\[ 2r = 2 \implies r = \frac{2}{2} = 1 \]

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

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