The area of the triangle formed by the tangents from (1,3) to the circle `x^(2)+y^(2)-4x+6y+1=0` and its chord of contact is

Options

A. `(250sqrt(3))/37`
B. `(125sqrt(3))/37`
C. `(250sqrt(3))/17`
D. `(125sqrt(3))/17`

Answer

To find the area of the triangle formed by the tangents from the point (1, 3) to the circle given by the equation \(x^2 + y^2 - 4x + 6y + 1 = 0\) and its chord of contact, we can follow these steps:

### Step 1: Convert the Circle Equation to Standard Form
The given equation of the circle is:
\[ x^2 + y^2 - 4x + 6y + 1 = 0 \]
We can rewrite it in standard form by completing the square.

1. Rearranging the equation:
\[ (x^2 - 4x) + (y^2 + 6y) + 1 = 0 \]

2. Completing the square for \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

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

4. Substituting back into the equation:
\[ (x - 2)^2 - 4 + (y + 3)^2 - 9 + 1 = 0 \]
\[ (x - 2)^2 + (y + 3)^2 - 12 = 0 \]
\[ (x - 2)^2 + (y + 3)^2 = 12 \]

The center of the circle is \((2, -3)\) and the radius \(R\) is \(\sqrt{12} = 2\sqrt{3}\).

### Step 2: Find the Equation of the Chord of Contact
The chord of contact from the point \((x_1, y_1) = (1, 3)\) is given by:
\[ xx_1 + yy_1 - g(x + x_1) - f(y + y_1) + c = 0 \]
Where \(g = -2\), \(f = 3\), and \(c = 1\).

Substituting the values:
\[ x(1) + y(3) - (-2)(x + 1) - 3(y + 3) + 1 = 0 \]
Simplifying:
\[ x + 3y + 2x + 2 - 3y - 9 + 1 = 0 \]
\[ 3x + 0y - 6 = 0 \implies x = 2 \]

### Step 3: Find the Length of the Perpendicular from the Center to the Chord
The distance \(OM\) from the center \(O(2, -3)\) to the line \(x = 2\) is simply the horizontal distance:
\[ OM = 0 \]

### Step 4: Find the Length of the Tangents
The length of the tangents from point \(P(1, 3)\) to the circle can be calculated using the formula:
\[ L = \sqrt{d^2 - R^2} \]
Where \(d\) is the distance from point \(P\) to the center \(O\):
\[ d = \sqrt{(1 - 2)^2 + (3 + 3)^2} = \sqrt{1 + 36} = \sqrt{37} \]
Thus:
\[ L = \sqrt{37 - 12} = \sqrt{25} = 5 \]

### Step 5: Calculate the Area of Triangle PQR
The area \(A\) of triangle \(PQR\) can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the length of the tangents (which is \(L\)) and the height is \(OM\):
\[ A = \frac{1}{2} \times 5 \times 0 = 0 \]

### Final Area Calculation
Since the height \(OM\) is 0, the area of triangle \(PQR\) is:
\[ \text{Area} = 0 \]