lf a circle `C` passing through `(4,0)` touches the circle `x^2 + y^2 + 4x-6y-12 = 0` externally at a point `(1, -1),` then the radius of the circle `C` is :-

Options

A. `sqrt(57)`
B. 4
C. `2sqrt(5)`
D. 5

Answer

To find the radius of the circle \( C \) that passes through the point \( (4, 0) \) and touches the circle defined by the equation \( x^2 + y^2 + 4x - 6y - 12 = 0 \) externally at the point \( (1, -1) \), we can follow these steps:

### Step 1: Rewrite the given circle's equation
The equation of the circle can be rewritten in standard form. Start with:
\[ x^2 + y^2 + 4x - 6y - 12 = 0 \]
Group the \( x \) and \( y \) terms:
\[ (x^2 + 4x) + (y^2 - 6y) = 12 \]
Now, complete the square for \( x \) and \( y \):
\[ (x^2 + 4x + 4) + (y^2 - 6y + 9) = 12 + 4 + 9 \]
This simplifies to:
\[ (x + 2)^2 + (y - 3)^2 = 25 \]
Thus, the center of the circle is \( (-2, 3) \) and the radius is \( 5 \).

### Step 2: Find the equation of the tangent line at the point of tangency
The point of tangency is \( (1, -1) \). The slope of the radius from the center \( (-2, 3) \) to the point \( (1, -1) \) is:
\[ \text{slope} = \frac{-1 - 3}{1 + 2} = \frac{-4}{3} \]
The slope of the tangent line, being perpendicular to the radius, is the negative reciprocal:
\[ \text{slope of tangent} = \frac{3}{4} \]
Using the point-slope form of the line equation:
\[ y - (-1) = \frac{3}{4}(x - 1) \]
This simplifies to:
\[ y + 1 = \frac{3}{4}x - \frac{3}{4} \]
\[ 3x - 4y - 7 = 0 \]
This is the equation of the tangent line.

### Step 3: Form the equation of circle \( C \)
The circle \( C \) can be expressed as:
\[ x^2 + y^2 + 4x - 6y - 12 + \lambda(3x - 4y - 7) = 0 \]
Substituting \( (4, 0) \) into this equation gives:
\[ 16 + 0 + 16 - 0 - 12 + \lambda(12 + 0 - 7) = 0 \]
Simplifying:
\[ 20 + 5\lambda = 0 \implies \lambda = -4 \]
Substituting \( \lambda \) back into the equation gives:
\[ x^2 + y^2 + 4x - 6y - 12 - 4(3x - 4y - 7) = 0 \]
This simplifies to:
\[ x^2 + y^2 - 8x + 10y + 16 = 0 \]

### Step 4: Identify the center and radius of circle \( C \)
The standard form of the circle is:
\[ (x - 4)^2 + (y + 5)^2 = 25 \]
From this, we see that the center is \( (4, -5) \) and the radius \( R \) is:
\[ R = \sqrt{16 + 25} = \sqrt{41} \]

### Step 5: Calculate the radius
The radius of circle \( C \) is:
\[ R = 5 \]

### Final Answer
The radius of circle \( C \) is \( 5 \).