The length of the common chord of the circles `x^(2)+y^(2)-2x-1=0` and `x^(2)+y^(2)+4y-1=0`, is

Options

A. `sqrt(15//2)`
B. `sqrt(15)`
C. `2sqrt(15)`
D. none of these

Answer

To find the length of the common chord of the circles given by the equations \( x^2 + y^2 - 2x - 1 = 0 \) and \( x^2 + y^2 + 4y - 1 = 0 \), we can follow these steps:

### Step 1: Rewrite the Circle Equations
First, we need to rewrite the given equations in standard form.

1. For the first circle:
\[ x^2 + y^2 - 2x - 1 = 0 \implies (x^2 - 2x + 1) + y^2 = 1 \implies (x-1)^2 + y^2 = 1 \]
This circle has center \( (1, 0) \) and radius \( r_1 = 1 \).

2. For the second circle:
\[ x^2 + y^2 + 4y - 1 = 0 \implies x^2 + (y^2 + 4y + 4) = 5 \implies x^2 + (y+2)^2 = 5 \]
This circle has center \( (0, -2) \) and radius \( r_2 = \sqrt{5} \).

### Step 2: Find the Equation of the Common Chord
The equation of the common chord can be found using the formula \( S - S_1 = 0 \), where \( S \) and \( S_1 \) are the equations of the two circles.

Subtracting the second circle's equation from the first:
\[ (x^2 + y^2 - 2x - 1) - (x^2 + y^2 + 4y - 1) = 0 \]
This simplifies to:
\[ -2x - 4y = 0 \implies 2x + 4y = 0 \implies x + 2y = 0 \]
So, the equation of the common chord is \( x + 2y = 0 \).

### Step 3: Find the Distance from the Center of One Circle to the Common Chord
We will find the distance \( d \) from the center of the first circle \( (1, 0) \) to the line \( x + 2y = 0 \).

The formula for the distance \( d \) from a point \( (x_1, y_1) \) to the line \( ax + by + c = 0 \) is:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]
Here, \( a = 1, b = 2, c = 0 \), and the point is \( (1, 0) \):
\[ d = \frac{|1 \cdot 1 + 2 \cdot 0 + 0|}{\sqrt{1^2 + 2^2}} = \frac{|1|}{\sqrt{1 + 4}} = \frac{1}{\sqrt{5}} \]

### Step 4: Calculate the Length of the Common Chord
The length \( L \) of the common chord is given by the formula:
\[ L = 2 \sqrt{r^2 - d^2} \]
Where \( r \) is the radius of the first circle and \( d \) is the distance calculated above.

Substituting the values:
- Radius \( r_1 = 1 \)
- Distance \( d = \frac{1}{\sqrt{5}} \)

Calculating \( d^2 \):
\[ d^2 = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5} \]

Now substituting into the length formula:
\[ L = 2 \sqrt{1^2 - \frac{1}{5}} = 2 \sqrt{1 - \frac{1}{5}} = 2 \sqrt{\frac{4}{5}} = 2 \cdot \frac{2}{\sqrt{5}} = \frac{4}{\sqrt{5}} = \frac{4\sqrt{5}}{5} \]

### Final Answer
The length of the common chord is \( \frac{4\sqrt{5}}{5} \).