The circles `x^2+y^2-6x-8y+12=0, x^2+y^2-4x+6y+k=0` , cut orthogonally then k=

Options

A. `-12`
B. `-24`
C. `12`
D. `24`

Answer

To solve the problem of finding the value of \( k \) such that the circles \( x^2 + y^2 - 6x - 8y + 12 = 0 \) and \( x^2 + y^2 - 4x + 6y + k = 0 \) cut orthogonally, we will follow these steps:

### Step 1: Rewrite the equations of the circles
The first circle is given by:
\[ x^2 + y^2 - 6x - 8y + 12 = 0 \]
We can rewrite this in the standard form \( (x - h)^2 + (y - k)^2 = r^2 \).

### Step 2: Identify the parameters of the first circle
From the equation, we can identify:
- \( g_1 = -3 \) (since \( 2g_1 = -6 \))
- \( f_1 = -4 \) (since \( 2f_1 = -8 \))
- \( c_1 = 12 \)

### Step 3: Rewrite the second circle's equation
The second circle is given by:
\[ x^2 + y^2 - 4x + 6y + k = 0 \]
Again, we will compare it with the general form.

### Step 4: Identify the parameters of the second circle
From the second equation, we can identify:
- \( g_2 = -2 \) (since \( 2g_2 = -4 \))
- \( f_2 = 3 \) (since \( 2f_2 = 6 \))
- \( c_2 = k \)

### Step 5: Use the orthogonality condition
The circles cut orthogonally if the following condition holds:
\[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
Substituting the values we found:
\[ 2(-3)(-2) + 2(-4)(3) = 12 + k \]

### Step 6: Simplify the equation
Calculating the left side:
\[ 2 \cdot 6 - 24 = 12 + k \]
\[ 12 - 24 = 12 + k \]
\[ -12 = 12 + k \]

### Step 7: Solve for \( k \)
Now, isolate \( k \):
\[ k = -12 - 12 \]
\[ k = -24 \]

Thus, the value of \( k \) is:
\[ \boxed{-24} \]