The length of the chord of contact of (-2,3) with respect to the circle `x^(2)+y^(2)-2x+4y+1=0` is

Options

A. `15sqrt(13/3)`
B. `5sqrt(3/13)`
C. `4sqrt(15/17)`
D. `15sqrt(3/13)`

Answer

To find the length of the chord of contact from the point (-2, 3) with respect to the circle given by the equation \(x^2 + y^2 - 2x + 4y + 1 = 0\), we will follow these steps:

### Step 1: Identify the center and radius of the circle
The general form of the circle's equation is \(x^2 + y^2 + 2gx + 2fy + c = 0\). From the given equation, we can identify:
- \(2g = -2 \Rightarrow g = -1\)
- \(2f = 4 \Rightarrow f = 2\)
- \(c = 1\)

The center \(C\) of the circle is given by \((-g, -f)\):
\[ C = (1, -2) \]

To find the radius \(r\), we use the formula:
\[ r = \sqrt{g^2 + f^2 - c} \]
Substituting the values:
\[ r = \sqrt{(-1)^2 + (2)^2 - 1} = \sqrt{1 + 4 - 1} = \sqrt{4} = 2 \]

### Step 2: Write the equation of the chord of contact
The chord of contact from the point \((x_1, y_1) = (-2, 3)\) to the circle is given by the equation:
\[ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \]
Substituting the values:
\[ x(-2) + y(3) - 1(x - 2) + 2(y + 3) + 1 = 0 \]
This simplifies to:
\[ -2x + 3y - 1(x + 2) + 2y + 6 + 1 = 0 \]
Combining like terms:
\[ -3x + 5y + 5 = 0 \]

### Step 3: Find the perpendicular distance from the center to the chord
The perpendicular distance \(d\) from the center \(C(1, -2)\) to the line \(-3x + 5y + 5 = 0\) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
where \(A = -3\), \(B = 5\), \(C = 5\), and \((x_0, y_0) = (1, -2)\):
\[ d = \frac{|-3(1) + 5(-2) + 5|}{\sqrt{(-3)^2 + 5^2}} = \frac{|-3 - 10 + 5|}{\sqrt{9 + 25}} = \frac{|-8|}{\sqrt{34}} = \frac{8}{\sqrt{34}} \]

### Step 4: Use Pythagorean theorem to find half the length of the chord
Let \(D\) be the foot of the perpendicular from \(C\) to the chord. The length of the chord \(AB\) can be found using the relation:
\[ AB = 2 \sqrt{r^2 - d^2} \]
Substituting the values:
\[ AB = 2 \sqrt{2^2 - \left(\frac{8}{\sqrt{34}}\right)^2} \]
Calculating \(d^2\):
\[ d^2 = \left(\frac{8}{\sqrt{34}}\right)^2 = \frac{64}{34} = \frac{32}{17} \]
Now substituting back:
\[ AB = 2 \sqrt{4 - \frac{32}{17}} = 2 \sqrt{\frac{68}{17} - \frac{32}{17}} = 2 \sqrt{\frac{36}{17}} = 2 \cdot \frac{6}{\sqrt{17}} = \frac{12}{\sqrt{17}} \]

### Final Answer
Thus, the length of the chord of contact is:
\[ \frac{12}{\sqrt{17}} \]