If length of the common chord of the circles `x^2 + y^2 + 2x + 3y + 1 = 0 and x^2 + y^2 + 4x + 3y + 2 = 0` then the value of `[a].` (where [ - ] denotes greatest integer function)

Options

A. (a)`(9)/(2)`
B. (b)`(3)/(2)`
C. (c)`3sqrt(2)`
D. (d)`2sqrt(2)`

Answer

To find the length of the common chord of the circles given by the equations \(x^2 + y^2 + 2x + 3y + 1 = 0\) and \(x^2 + y^2 + 4x + 3y + 2 = 0\), we will follow these steps:

### Step 1: Rewrite the equations of the circles in standard form

The general equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.

For the first circle:
\[ x^2 + y^2 + 2x + 3y + 1 = 0 \]
Rearranging gives:
\[ x^2 + 2x + y^2 + 3y + 1 = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x + 1)^2 - 1 + (y + \frac{3}{2})^2 - \frac{9}{4} + 1 = 0 \]
\[ (x + 1)^2 + (y + \frac{3}{2})^2 = \frac{9}{4} \]
Thus, the center \(C_1\) is \((-1, -\frac{3}{2})\) and the radius \(r_1 = \frac{3}{2}\).

For the second circle:
\[ x^2 + y^2 + 4x + 3y + 2 = 0 \]
Rearranging gives:
\[ x^2 + 4x + y^2 + 3y + 2 = 0 \]
Completing the square:
\[ (x + 2)^2 - 4 + (y + \frac{3}{2})^2 - \frac{9}{4} + 2 = 0 \]
\[ (x + 2)^2 + (y + \frac{3}{2})^2 = \frac{9}{4} \]
Thus, the center \(C_2\) is \((-2, -\frac{3}{2})\) and the radius \(r_2 = \frac{3}{2}\).

### Step 2: Find the distance between the centers \(C_1\) and \(C_2\)

The distance \(d\) between the centers \(C_1(-1, -\frac{3}{2})\) and \(C_2(-2, -\frac{3}{2})\) is given by:
\[ d = \sqrt{((-1) - (-2))^2 + \left(-\frac{3}{2} - (-\frac{3}{2})\right)^2} \]
\[ d = \sqrt{(1)^2 + (0)^2} = 1 \]

### Step 3: Use the formula for the length of the common chord

The length of the common chord \(L\) of two intersecting circles can be calculated using the formula:
\[ L = \sqrt{(r_1^2 + r_2^2 - d^2)} \]
Substituting the values:
\[ L = \sqrt{\left(\left(\frac{3}{2}\right)^2 + \left(\frac{3}{2}\right)^2 - 1^2\right)} \]
\[ L = \sqrt{\left(\frac{9}{4} + \frac{9}{4} - 1\right)} = \sqrt{\left(\frac{18}{4} - \frac{4}{4}\right)} = \sqrt{\left(\frac{14}{4}\right)} = \sqrt{\frac{7}{2}} \]

### Step 4: Calculate the value of \([L]\)

Now, we need to find the greatest integer less than or equal to \(L\):
\[ L = \sqrt{\frac{7}{2}} \approx \sqrt{3.5} \approx 1.87 \]
Thus, the greatest integer function \([L] = 1\).

### Conclusion

The value of \([L]\) is:
\[ \boxed{1} \]