The common chord of the circle x ^ 2 + y ^ 2 + 4x + 1 = 0 and x ^ 2 + y ^ 2 + 6x + 2y + 3 = 0 is. (a) x + y - 1 = 0 (b) x + y - 2 = 0 (c) x + y + 1 = 0 (d) x + y + 2 = 0

To find the common chord of the two given circles, we start by writing down their equations:

1. **Circle 1**: \( x^2 + y^2 + 4x + 1 = 0 \)
2. **Circle 2**: \( x^2 + y^2 + 6x + 2y + 3 = 0 \)

### Step 1: Standard Form of the Circle Equations

Let's rewrite each equation in the standard form \((x - h)^2 + (y - k)^2 = r^2\).

**For Circle 1**: 

\[
x^2 + y^2 + 4x + 1 = 0
\]

Complete the square for the \(x\) terms:

\[
x^2 + 4x = (x+2)^2 - 4
\]

So, the equation becomes:

\[
(x+2)^2 + y^2 = 4 - 1
\]

\[
(x+2)^2 + y^2 = -1
\]

Since the radius squared is negative, this is not a valid circle. However, since we are only interested in the common chord, this does not affect the following calculations for finding the line of intersection.

**For Circle 2**: 

\[
x^2 + y^2 + 6x + 2y + 3 = 0
\]

Complete the square for the \(x\) and \(y\) terms:

\[
x^2 + 6x = (x+3)^2 - 9
\]
\[
y^2 + 2y = (y+1)^2 - 1
\]

Substitute back into the circle equation:

\[
(x+3)^2 - 9 + (y+1)^2 - 1 = 0
\]

\[
(x+3)^2 + (y+1)^2 = 10
\]

This is a valid circle with center \((-3, -1)\) and radius \(\sqrt{10}\).

### Step 2: Finding the Common Chord

To find the common chord of two circles, we subtract their equations.

Given:

1. \( x^2 + y^2 + 4x + 1 = 0 \)
2. \( x^2 + y^2 + 6x + 2y + 3 = 0 \)

Subtract equation 1 from equation 2:

\[
(x^2 + y^2 + 6x + 2y + 3) - (x^2 + y^2 + 4x + 1) = 0
\]

Simplify:

\[
6x + 2y + 3 - 4x - 1 = 0
\]

\[
2x + 2y + 2 = 0
\]

Divide by 2:

\[
x + y + 1 = 0
\]

### Answer

The equation of the common chord is \(\boxed{x + y + 1 = 0}\).

So, the correct option is (c) \( x + y + 1 = 0 \).

Would you like more details, or do you have any questions?

Here are five related questions to expand on this topic:

1. What is the geometric interpretation of the common chord of two circles?
2. How would you find the points of intersection of two circles?
3. What if the common chord is parallel to one of the coordinate axes?
4. Can two circles have more than one common chord?
5. How does the distance between the centers of two circles affect the common chord?

**Tip:** The common chord of two intersecting circles is the line joining the points where the circles intersect. It can be found by subtracting the equations of the two circles.

Discover how to find the common chord of two circles, x^2 + y^2 + 4x + 1 = 0 and x^2 + y^2 + 6x + 2y + 3 = 0, through step-by-step solution. Learn circle equations and completing the square method in algebra. Suitable for Grades 10-12.

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