Find the length of the common chord of the parabola `y^2=4(x+3)` and the circle `x^2+y^2+4x=0` .

Explanation

To find the length of the common chord of the parabola \( y^2 = 4(x + 3) \) and the circle \( x^2 + y^2 + 4x = 0 \), we can follow these steps:

### Step 1: Rewrite the equations
First, we rewrite the equations of the parabola and the circle in a more usable form.

**Parabola:**
\[ y^2 = 4(x + 3) \implies y^2 = 4x + 12 \]

**Circle:**
\[ x^2 + y^2 + 4x = 0 \implies x^2 + 4x + y^2 = 0 \implies (x + 2)^2 + y^2 = 4 \]
This shows that the circle is centered at (-2, 0) with a radius of 2.

### Step 2: Substitute the parabola equation into the circle equation
We substitute \( y^2 = 4x + 12 \) into the circle's equation:
\[ x^2 + (4x + 12) + 4x = 0 \]
This simplifies to:
\[ x^2 + 8x + 12 = 0 \]

### Step 3: Solve the quadratic equation
Now we will solve the quadratic equation \( x^2 + 8x + 12 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 8, c = 12 \):
\[ x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1} = \frac{-8 \pm \sqrt{64 - 48}}{2} = \frac{-8 \pm \sqrt{16}}{2} \]
\[ x = \frac{-8 \pm 4}{2} \]
Calculating the two roots:
\[ x_1 = \frac{-8 + 4}{2} = -2 \quad \text{and} \quad x_2 = \frac{-8 - 4}{2} = -6 \]

### Step 4: Find the corresponding y-coordinates
Now we find the y-coordinates for both x-values using the parabola equation \( y^2 = 4x + 12 \).

For \( x = -2 \):
\[ y^2 = 4(-2) + 12 = -8 + 12 = 4 \implies y = \pm 2 \]
So the points are \( (-2, 2) \) and \( (-2, -2) \).

For \( x = -6 \):
\[ y^2 = 4(-6) + 12 = -24 + 12 = -12 \]
This does not yield real values for y, confirming that the points \( (-2, 2) \) and \( (-2, -2) \) are the only intersection points.

### Step 5: Calculate the length of the common chord
The length of the common chord can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates \( A(-2, 2) \) and \( B(-2, -2) \):
\[ AB = \sqrt{(-2 - (-2))^2 + (2 - (-2))^2} = \sqrt{0^2 + (2 + 2)^2} = \sqrt{0 + 4^2} = \sqrt{16} = 4 \]

### Final Answer
The length of the common chord is \( 4 \).

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