If the circles `x^2+y^2-2x-2y-7=0` and `x^2+y^2+4x+2y+k=0` cut orthogonally, then the length of the common chord of the circles is `a.(12)/(sqrt(13))` b. 2 c. 5 d. 8

Answer

To solve the problem, we need to find the length of the common chord of the two circles given that they cut orthogonally. Let's go through the steps systematically.

### Step 1: Identify the equations of the circles
The equations of the circles are:
1. \( C_1: x^2 + y^2 - 2x - 2y - 7 = 0 \)
2. \( C_2: x^2 + y^2 + 4x + 2y + k = 0 \)

### Step 2: Rewrite the equations in standard form
We can rewrite these equations in standard form by completing the square.

For \( C_1 \):
\[ x^2 - 2x + y^2 - 2y = 7 \]
Completing the square:
\[ (x-1)^2 + (y-1)^2 = 9 \]
This gives us a circle with center \( (1, 1) \) and radius \( r_1 = 3 \).

For \( C_2 \):
\[ x^2 + 4x + y^2 + 2y = -k \]
Completing the square:
\[ (x+2)^2 + (y+1)^2 = k + 5 \]
This gives us a circle with center \( (-2, -1) \) and radius \( r_2 = \sqrt{k + 5} \).

### Step 3: Use the condition for orthogonality
The condition for two circles to cut orthogonally is given by:
\[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
Where \( g_1, f_1, c_1 \) are from the first circle and \( g_2, f_2, c_2 \) are from the second circle.

From \( C_1 \):
- \( g_1 = -1 \)
- \( f_1 = -1 \)
- \( c_1 = -7 \)

From \( C_2 \):
- \( g_2 = 2 \)
- \( f_2 = 1 \)
- \( c_2 = k \)

Substituting these values into the orthogonality condition:
\[ 2(-1)(2) + 2(-1)(1) = -7 + k \]
This simplifies to:
\[ -4 - 2 = -7 + k \]
\[ -6 = -7 + k \]
Thus, we find:
\[ k = 1 \]

### Step 4: Find the equation of the common chord
The equation of the common chord can be found using:
\[ s_1 - s_2 = 0 \]
Substituting \( k = 1 \) into \( C_2 \):
\[ s_1: x^2 + y^2 - 2x - 2y - 7 = 0 \]
\[ s_2: x^2 + y^2 + 4x + 2y + 1 = 0 \]
Subtracting these gives:
\[ -6x - 4y - 8 = 0 \]
This simplifies to:
\[ 3x + 2y + 4 = 0 \]

### Step 5: Find the distance from the center of \( C_1 \) to the common chord
The center of \( C_1 \) is \( (1, 1) \). The distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our line \( 3x + 2y + 4 = 0 \):
- \( A = 3, B = 2, C = 4 \)
- Point \( (x_0, y_0) = (1, 1) \)

Calculating the distance:
\[ d = \frac{|3(1) + 2(1) + 4|}{\sqrt{3^2 + 2^2}} = \frac{|3 + 2 + 4|}{\sqrt{9 + 4}} = \frac{9}{\sqrt{13}} = \frac{9}{\sqrt{13}} \]

### Step 6: Find the length of the common chord
The length of the common chord \( AB \) is given by:
\[ AB = 2 \sqrt{r^2 - d^2} \]
Where \( r = 3 \) (radius of \( C_1 \)) and \( d = \frac{9}{\sqrt{13}} \).

Calculating \( d^2 \):
\[ d^2 = \left(\frac{9}{\sqrt{13}}\right)^2 = \frac{81}{13} \]
Now, calculate \( r^2 \):
\[ r^2 = 3^2 = 9 \]
Thus:
\[ AB = 2 \sqrt{9 - \frac{81}{13}} = 2 \sqrt{\frac{117 - 81}{13}} = 2 \sqrt{\frac{36}{13}} = 2 \cdot \frac{6}{\sqrt{13}} = \frac{12}{\sqrt{13}} \]

### Final Answer
The length of the common chord is \( \frac{12}{\sqrt{13}} \).