Find k given that x2+y2−12x+8y+k=0 with a radius of 4 units.

We are given the equation of a circle in general form:

\[
x^2 + y^2 - 12x + 8y + k = 0
\]

and we know that the radius of the circle is 4 units. Our goal is to find the value of \( k \).

### Step 1: Convert the equation to standard form

To do this, we'll complete the square for both \( x \) and \( y \).

1. **For \( x \):**

   \[
   x^2 - 12x
   \]

   To complete the square, add and subtract \( \left(\frac{-12}{2}\right)^2 = 36 \):

   \[
   x^2 - 12x + 36 - 36
   \]

   This becomes:

   \[
   (x - 6)^2 - 36
   \]

2. **For \( y \):**

   \[
   y^2 + 8y
   \]

   To complete the square, add and subtract \( \left(\frac{8}{2}\right)^2 = 16 \):

   \[
   y^2 + 8y + 16 - 16
   \]

   This becomes:

   \[
   (y + 4)^2 - 16
   \]

### Step 2: Substitute back into the original equation

Substitute the completed squares into the original equation:

\[
(x - 6)^2 - 36 + (y + 4)^2 - 16 + k = 0
\]

Combine constants:

\[
(x - 6)^2 + (y + 4)^2 - 52 + k = 0
\]

This simplifies to:

\[
(x - 6)^2 + (y + 4)^2 = 52 - k
\]

### Step 3: Use the radius to solve for \( k \)

The equation of a circle in standard form is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Here, \( r = 4 \), so \( r^2 = 16 \).

Thus, we have:

\[
52 - k = 16
\]

Solve for \( k \):

\[
k = 52 - 16 = 36
\]

### Final Answer

The value of \( k \) is \( \boxed{36} \).

Would you like more details on this problem or have any questions?

Here are some related questions you might find interesting:

1. How do you complete the square for a general quadratic equation?
2. What is the relationship between the general form and the standard form of a circle's equation?
3. How can you derive the radius of a circle from its equation?
4. How do you find the center of a circle given its general equation?
5. What happens to the equation if the radius changes?

**Tip:** Always double-check your algebraic manipulations when completing the square, as small errors can lead to incorrect results.

Learn how to find the value of k in the circle equation x^2 + y^2 - 12x + 8y + k = 0 with a given radius of 4 units. This problem involves completing the square and applying properties of circle equations. Suitable for students in Grades 9-10.

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