Find k if the following pairs of circles are orthogonal
`x^2+y^2-6x-8y+12=0`
`x^2+y^2-4x+6y+k=0`

Answer

To find the value of \( k \) such that the given circles are orthogonal, we will follow these steps:

### Step 1: Write the equations of the circles in standard form
The given equations of the circles are:
1. \( x^2 + y^2 - 6x - 8y + 12 = 0 \)
2. \( x^2 + y^2 - 4x + 6y + k = 0 \)

### Step 2: Identify coefficients from the general form of the circle equation
The general form of a circle is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the first circle, we can identify:
- \( g_1 = -3 \) (since \( 2g_1 = -6 \))
- \( f_1 = -4 \) (since \( 2f_1 = -8 \))
- \( c_1 = 12 \)

From the second circle, we can identify:
- \( g_2 = -2 \) (since \( 2g_2 = -4 \))
- \( f_2 = 3 \) (since \( 2f_2 = 6 \))
- \( c_2 = k \)

### Step 3: Use the condition for orthogonality of circles
The circles are orthogonal if the following condition holds:
\[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
Substituting the identified values into this equation:
\[ 2(-3)(-2) + 2(-4)(3) = 12 + k \]

### Step 4: Simplify the left-hand side
Calculating the left-hand side:
\[ 2 \cdot 6 - 24 = 12 + k \]
This simplifies to:
\[ 12 - 24 = 12 + k \]
\[ -12 = 12 + k \]

### Step 5: Solve for \( k \)
Now, we can isolate \( k \):
\[ k = -12 - 12 \]
\[ k = -24 \]

### Final Answer
Thus, the value of \( k \) is \( -24 \).

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